Continuity of a Function

Continuity at a Point: Formal Definition

The concept of continuity is fundamental in calculus and describes functions whose graphs can be drawn without lifting the pen. A function that is continuous at a particular point has a predictable behavior near that point, where the function's value precisely matches the limit it approaches.

Intuitive Understanding of Continuity

Intuitively, a function $f(x)$ is continuous at a point $x = a$ if there are no abrupt changes or breaks in its graph at that point. If you imagine tracing the graph of the function with a pen, you should be able to pass through the point $(a, f(a))$ without lifting the pen from the paper. This implies the graph is smooth and unbroken at $x=a$.

Conversely, a function is discontinuous at a point if its graph has a gap, a jump, a hole, or behaves erratically near that point. These are points where the function's behavior is unpredictable or its value doesn't "fit" with the values around it.

The formal definition precisely captures this intuitive idea using the concept of limits.

Formal Definition of Continuity at a Point

A function $f(x)$ is defined to be continuous at a point $x = a$ if and only if all three of the following conditions are satisfied:

1. The function is defined at $a$ ($f(a)$ exists): The point $a$ must be in the domain of the function $f(x)$. This means that when you plug $x=a$ into the function rule, you get a specific, finite real number as the output, $f(a)$. The point $(a, f(a))$ must be a point on the graph.

2. The limit of the function as $x$ approaches $a$ exists ($\lim\limits_{x \to a} f(x)$ exists): The limit of $f(x)$ as $x$ approaches $a$ must exist as a finite real number. Recall from the concept of limits that for the bilateral limit $\lim\limits_{x \to a} f(x)$ to exist, both the left hand limit (LHL) and the right hand limit (RHL) must exist, be finite, and be equal to the same value $L$. That is:

   $\lim\limits_{x \to a^-} f(x) = \lim\limits_{x \to a^+} f(x) = L$, where $L$ is a finite real number.

3. The value of the limit is equal to the value of the function at $a$ ($\lim\limits_{x \to a} f(x) = f(a)$): This is the crucial condition that ties the function's behavior near $a$ (the limit) to its actual value at $a$. It means that the value the function is approaching as $x$ gets close to $a$ is exactly the value it hits when $x$ is equal to $a$. If the limit exists ($=L$) and the function value is defined ($=f(a)$), this condition requires $L = f(a)$.

If even one of these three conditions is not met, the function $f(x)$ is said to be discontinuous at the point $x = a$. Understanding which condition fails helps in classifying the type of discontinuity.

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Continuity in an Interval (Open and Closed)

The concept of continuity at a single point can be extended to define what it means for a function to be continuous over an entire interval of real numbers. This involves ensuring the function is continuous at every point within that interval, with special considerations for the endpoints in the case of a closed interval.

Continuity on an Open Interval

A function $f(x)$ is said to be continuous on an open interval $(a, b)$ if it is continuous at every single point $c$ within that interval. That is, for every $c$ such that $a < c < b$, the three conditions for continuity at the point $c$ must be satisfied:

1. $f(c)$ is defined.
2. $\lim\limits_{x \to c} f(x)$ exists.
3. $\lim\limits_{x \to c} f(x) = f(c)$.

If a function is continuous on $(a, b)$, its graph will be an unbroken curve between the vertical lines $x=a$ and $x=b$, although its behavior at $a$ and $b$ themselves is not included in this definition.

Continuity on a Closed Interval

Defining continuity on a closed interval $[a, b]$ requires considering the endpoints separately, as the full bilateral limit $\lim\limits_{x \to a} f(x)$ or $\lim\limits_{x \to b} f(x)$ might not make sense (e.g., if the function's domain is restricted to the interval). Instead, we use one-sided limits at the endpoints.

A function $f(x)$ is said to be continuous on a closed interval $[a, b]$ if the following three conditions are met:

1. $f$ is continuous on the open interval $(a, b)$: The function must be continuous at every point $c$ such that $a < c < b$. This means the graph is unbroken *within* the interval.

2. $f$ is continuous from the right at $a$: At the left endpoint $a$, the function's value must equal the limit as $x$ approaches $a$ from within the interval (i.e., from the right side). This means:

   $\lim\limits_{x \to a^+} f(x) = f(a)$

   Note that $f(a)$ must also be defined for this condition to be met.

3. $f$ is continuous from the left at $b$: At the right endpoint $b$, the function's value must equal the limit as $x$ approaches $b$ from within the interval (i.e., from the left side). This means:

   $\lim\limits_{x \to b^-} f(x) = f(b)$

   Note that $f(b)$ must also be defined for this condition to be met.

For continuity on a closed interval $[a, b]$, the graph must be an unbroken curve from the point $(a, f(a))$ to $(b, f(b))$. The conditions ensure that the function approaches the correct values at the endpoints from within the interval.

Examples of Functions Continuous on Their Domains

Many elementary functions encountered in calculus are continuous on their natural domains. Recognizing these allows us to quickly identify intervals of continuity.

• Polynomial Functions: E.g., $f(x) = x^2 - 3x + 5$. Continuous on $(-\infty, \infty)$.
• Rational Functions: E.g., $f(x) = \frac{x+1}{x-2}$. Continuous on their domain, which is all real numbers except where the denominator is zero. For this example, continuous on $(-\infty, 2) \cup (2, \infty)$. It is discontinuous at $x=2$.
• Trigonometric Functions:
  • $\sin x$, $\cos x$: Continuous on $(-\infty, \infty)$.
  • $\tan x = \frac{\sin x}{\cos x}$: Continuous on its domain, where $\cos x \neq 0$, i.e., $x \neq \frac{\pi}{2} + n\pi$ for integer $n$. Continuous on intervals like $(-\pi/2, \pi/2), (\pi/2, 3\pi/2)$, etc.
  • $\cot x = \frac{\cos x}{\sin x}$: Continuous on its domain, where $\sin x \neq 0$, i.e., $x \neq n\pi$ for integer $n$. Continuous on intervals like $(0, \pi), (\pi, 2\pi)$, etc.
  • $\sec x = \frac{1}{\cos x}$: Continuous on the same intervals as $\tan x$.
  • $\text{cosec } x = \frac{1}{\sin x}$: Continuous on the same intervals as $\cot x$.
• Exponential Functions: $f(x) = a^x$ (for $a>0, a\neq 1$), $f(x) = e^x$. Continuous on $(-\infty, \infty)$.
• Logarithmic Functions: $f(x) = \log_a x$ (for $a>0, a\neq 1$), $f(x) = \ln x$. Continuous on their domain, which is $(0, \infty)$.
• Root Functions: $f(x) = \sqrt[n]{x}$.
  • If $n$ is even (e.g., $\sqrt{x}$): Continuous on its domain $[0, \infty)$.
  • If $n$ is odd (e.g., $\sqrt[3]{x}$): Continuous on its domain $(-\infty, \infty)$.
• Absolute Value Function: $f(x) = |x|$. Continuous on $(-\infty, \infty)$.

Also, sums, differences, products, and compositions of continuous functions are continuous on their respective domains. Quotients of continuous functions are continuous wherever the denominator is non-zero.

Types of Discontinuity

When a function $f(x)$ fails to be continuous at a point $x=a$ (meaning at least one of the three conditions for continuity at that point is not met), the function is said to have a discontinuity at $x=a$. Discontinuities are broadly classified into types based on the behavior of the function and its limit near the point of discontinuity.

Classification of Discontinuities

Discontinuities can be primarily categorized into two main types: Removable and Non-Removable.

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Properties of Continuous Functions (Algebra of Continuous Functions)

Just as limits have algebraic properties, functions that are continuous at a point also exhibit predictable behavior under standard arithmetic operations. These properties are directly derived from the corresponding limit properties and allow us to determine the continuity of functions that are built up from simpler continuous functions.

Theorems on Algebra of Continuous Functions

Let $f$ and $g$ be two real-valued functions that are both continuous at a point $x = a$. Based on the definition of continuity and the algebra of limits, the following combined functions are also continuous at $x = a$:

1. Sum Rule: The function $(f + g)(x) = f(x) + g(x)$ is continuous at $x = a$.

   Proof Idea: To show continuity of $(f+g)$ at $a$, we need to verify the three conditions. Since $f$ and $g$ are continuous at $a$, $f(a)$ and $g(a)$ are defined, so $(f+g)(a) = f(a) + g(a)$ is defined. Also, by the Sum Rule for limits, $\lim\limits_{x \to a} (f(x) + g(x)) = \lim\limits_{x \to a} f(x) + \lim\limits_{x \to a} g(x)$. Since $f$ and $g$ are continuous at $a$, we have $\lim\limits_{x \to a} f(x) = f(a)$ and $\lim\limits_{x \to a} g(x) = g(a)$. Therefore, $\lim\limits_{x \to a} (f(x) + g(x)) = f(a) + g(a) = (f+g)(a)$. The limit exists and equals the function value at $a$, so $(f+g)$ is continuous at $a$.

2. Difference Rule: The function $(f - g)(x) = f(x) - g(x)$ is continuous at $x = a$.

   Proof Idea: Similar to the sum rule, using the Difference Rule for limits: $\lim\limits_{x \to a} (f(x) - g(x)) = \lim\limits_{x \to a} f(x) - \lim\limits_{x \to a} g(x) = f(a) - g(a) = (f-g)(a)$.

3. Constant Multiple Rule: The function $(k \cdot f)(x) = k \cdot f(x)$ is continuous at $x = a$ for any real constant $k$.

   Proof Idea: Using the Constant Multiple Rule for limits: $\lim\limits_{x \to a} (k \cdot f(x)) = k \cdot \lim\limits_{x \to a} f(x) = k \cdot f(a) = (k \cdot f)(a)$.

4. Product Rule: The function $(f \cdot g)(x) = f(x) \cdot g(x)$ is continuous at $x = a$.

   Proof Idea: Using the Product Rule for limits: $\lim\limits_{x \to a} (f(x) \cdot g(x)) = \left(\lim\limits_{x \to a} f(x)\right) \cdot \left(\lim\limits_{x \to a} g(x)\right) = f(a) \cdot g(a) = (f \cdot g)(a)$.

5. Quotient Rule: The function $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ is continuous at $x = a$, provided $g(a) \neq 0$.

   Proof Idea: Using the Quotient Rule for limits: $\lim\limits_{x \to a} \left(\frac{f(x)}{g(x)}\right) = \frac{\lim\limits_{x \to a} f(x)}{\lim\limits_{x \to a} g(x)}$. Since $f$ and $g$ are continuous at $a$ and $g(a) \neq 0$, the limit of the denominator is $\lim\limits_{x \to a} g(x) = g(a) \neq 0$. So, the Quotient Rule applies: $\frac{\lim\limits_{x \to a} f(x)}{\lim\limits_{x \to a} g(x)} = \frac{f(a)}{g(a)} = \left(\frac{f}{g}\right)(a)$. The limit exists and equals the function value, so $(f/g)$ is continuous at $a$ (provided $g(a) \neq 0$). If $g(a) = 0$, the quotient function is undefined at $a$, failing the first condition of continuity.

These theorems can be extended by induction to the sum, difference, and product of any finite number of functions continuous at a point.

Implications for Common Functions

These properties are incredibly powerful because they allow us to conclude the continuity of many common functions simply by recognizing them as combinations of basic continuous functions:

• Consider the identity function $i(x) = x$. We know $\lim\limits_{x \to a} x = a$ and $i(a) = a$. So, $i(x)=x$ is continuous everywhere.
• Consider a constant function $c(x) = k$. We know $\lim\limits_{x \to a} k = k$ and $c(a) = k$. So, $c(x)=k$ is continuous everywhere.
• Using the Product Rule repeatedly, $x^n = x \cdot x \cdot \dots \cdot x$ (n times) is continuous everywhere for any positive integer $n$.
• Using the Constant Multiple Rule and Sum/Difference Rules, any polynomial $P(x) = c_n x^n + \dots + c_1 x + c_0$ (which is a sum/difference of constant multiples of powers of $x$) is continuous everywhere, i.e., on the interval $(-\infty, \infty)$.
• Using the Quotient Rule, any rational function $R(x) = \frac{P(x)}{Q(x)}$ (where $P, Q$ are polynomials) is continuous at every point $a$ where the denominator $Q(a) \neq 0$. This means rational functions are continuous on their entire domain.
• More generally, functions formed by adding, subtracting, multiplying, or dividing any functions known to be continuous (like trigonometric, exponential, logarithmic functions on their domains) will be continuous on the common part of their domains, excluding points where division by zero occurs.

Continuity of Composite Functions

Another fundamental way to combine functions is through composition. The continuity property also holds for the composition of continuous functions, which is a crucial theorem for analyzing the continuity of complex function expressions.

Theorem on Continuity of Composite Functions

Let $f$ and $g$ be two functions such that the composite function $(f \circ g)(x) = f(g(x))$ is defined in an interval containing $a$.

If the function $g(x)$ is continuous at the point $x = a$,

AND the function $f(u)$ is continuous at the point $u = g(a)$ (where $u$ is the output of $g$),

Then, the composite function $(f \circ g)(x) = f(g(x))$ is continuous at the point $x = a$.

Intuitive Understanding

The theorem states that if $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then the overall process of applying $g$ and then $f$ results in a continuous function at $a$.

Think of it this way: If $x$ is taken very close to $a$, because $g$ is continuous at $a$, the output $g(x)$ will be very close to $g(a)$. Now, since the input to $f$ ($u = g(x)$) is very close to $g(a)$, and $f$ is continuous at $g(a)$, the output $f(g(x))$ will be very close to $f(g(a))$. This precisely matches the condition for $(f \circ g)(x)$ to be continuous at $a$, which is $\lim\limits_{x \to a} f(g(x)) = f(g(a))$.

Using limit notation, the continuity of $g$ at $a$ means $\lim\limits_{x \to a} g(x) = g(a)$. The continuity of $f$ at $g(a)$ means $\lim\limits_{u \to g(a)} f(u) = f(g(a))$. The theorem implies that:

This property allows us to "pass the limit inside" a continuous function.

Significance and Applications

This theorem is exceptionally useful in calculus because it vastly simplifies the process of determining the continuity of many functions. Many complex functions can be viewed as compositions of simpler, known continuous functions.

• Example 1: Consider the function $h(x) = \sin(x^2)$.

  Let $g(x) = x^2$ and $f(u) = \sin u$. Then $h(x) = f(g(x))$.

  $g(x) = x^2$ is a polynomial function, which is continuous for all real numbers $x$.

  $f(u) = \sin u$ is a trigonometric function, which is continuous for all real numbers $u$.

  Since $g(x)$ is continuous everywhere, for any real $a$, $g(x)$ is continuous at $x=a$. The value $u=g(a)=a^2$ is a real number, and $f(u)=\sin u$ is continuous for all real numbers, including $u=a^2=g(a)$.

  By the theorem, $h(x) = f(g(x)) = \sin(x^2)$ is continuous at every point $x=a$. Therefore, $\sin(x^2)$ is continuous on $(-\infty, \infty)$.

• Example 2: Consider the function $h(x) = \sqrt{x^2+1}$.

  Let $g(x) = x^2+1$ and $f(u) = \sqrt{u}$. Then $h(x) = f(g(x))$.

  $g(x) = x^2+1$ is a polynomial function, continuous for all real numbers $x$.

  $f(u) = \sqrt{u}$ is a square root function, continuous on its domain $u \ge 0$.

  For the composition $f(g(x))$ to be continuous at $x=a$, $g(x)$ must be continuous at $a$ (which it is, everywhere) and $f(u)$ must be continuous at $u=g(a)$. This requires $g(a) \ge 0$.

  Let's check the condition $g(x) \ge 0$: $x^2+1 \ge 0$. This inequality is true for all real numbers $x$ since $x^2 \ge 0$ implies $x^2+1 \ge 1$. Thus, $g(x)$ is always in the domain of $f(u)$, and $f(u)$ is continuous at $u=g(x)$ for all real $x$.

  By the theorem, $h(x) = \sqrt{x^2+1}$ is continuous at every point $x=a$. Therefore, $\sqrt{x^2+1}$ is continuous on $(-\infty, \infty)$.

• Example 3: Consider the function $h(x) = e^{\cos x}$.

  Let $g(x) = \cos x$ and $f(u) = e^u$. Then $h(x) = f(g(x))$.

  $g(x) = \cos x$ is a trigonometric function, continuous for all real numbers $x$.

  $f(u) = e^u$ is an exponential function, continuous for all real numbers $u$.

  Since $g(x)$ is continuous everywhere, for any real $a$, $g(x)$ is continuous at $x=a$. The value $u=g(a)=\cos a$ is a real number, and $f(u)=e^u$ is continuous for all real numbers, including $u=\cos a=g(a)$.

  By the theorem, $h(x) = f(g(x)) = e^{\cos x}$ is continuous at every point $x=a$. Therefore, $e^{\cos x}$ is continuous on $(-\infty, \infty)$.

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Concepts of Limits and Continuity of a Function (Consolidated Applied)

As established earlier, continuity at a point is a specific condition defined using the concept of limits. A function $f(x)$ is continuous at $x=a$ if and only if the limit of $f(x)$ as $x$ approaches $a$ exists, the function is defined at $a$, and the limit value is equal to the function value at $a$. Applying these concepts together is key to solving problems involving piecewise functions or determining values that make a function continuous.

Specifically, for continuity at $x=a$, we require:

When dealing with piecewise functions, the points where the definition changes are potential points of discontinuity. To check for continuity at such a point, we evaluate the left hand limit (using the function definition for $x < a$), the right hand limit (using the function definition for $x > a$), and the function value at $a$ (using the definition for $x=a$), and then verify if they are all equal.

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