Continuity

Continuity is an important property of functions. Intuitively, it means the graph of the function has no breaks.

Continuity of Functions

Definition: Let $y = f(x)$ be defined in a neighborhood of $x_0$ . If $\lim_{x \to x_0} f(x) = f(x_0)$ , then $f(x)$ is said to be continuous at $x_0$ .

Types of Discontinuities

If a function is not continuous at a point, that point is called a discontinuity.

First Kind Discontinuity: Both left and right limits exist.
- Removable Discontinuity: $\lim_{x \to x_0} f(x)$ exists but does not equal $f(x_0)$ (or $f(x_0)$ is undefined).
- Jump Discontinuity: Left and right limits exist but are not equal.
Second Kind Discontinuity: At least one of the left or right limits does not exist.

Continuity of Elementary Functions

Basic Conclusion: All elementary functions are continuous on their domains.

Properties of Continuous Functions on Closed Intervals

Boundedness Theorem: A function continuous on a closed interval is bounded.
Extreme Value Theorem: A function continuous on a closed interval attains its maximum and minimum values.
Intermediate Value Theorem: If $f(x)$ is continuous on $[a, b]$ and $f(a) \neq f(b)$ , then for any $C$ between $f(a)$ and $f(b)$ , there exists $\xi \in (a, b)$ such that $f(\xi) = C$ .
Zero Point Theorem: If $f(x)$ is continuous on $[a, b]$ and $f(a) \cdot f(b) < 0$ , then there exists $\xi \in (a, b)$ such that $f(\xi) = 0$ .

Exercises

Determine whether $f(x) = \frac{1}{x}$ is continuous at $x = 0$ , and explain why.
Determine the continuity of $f(x) = \begin{cases} x^2, & x \leq 1 \\ 2x - 1, & x > 1 \end{cases}$ at $x = 1$ , and specify the type of discontinuity (if any).
Let $f(x)$ be continuous on $[0, 2]$ with $f(0) = -1, f(2) = 3$ . Prove that there exists $x_0$ in $(0, 2)$ such that $f(x_0) = 0$ .
Determine the continuity of $f(x) = \sin x$ on $\mathbb{R}$ .
If $f(x)$ is continuous on $[a, b]$ , does it have to be bounded and attain its maximum and minimum values?

Reference Answers

1. Is $f(x) = \frac{1}{x}$ continuous at $x = 0$ ? Explain why.

Solution: $f(x)$ is not defined at $x = 0$ , so it is not continuous there.

Answer: Not continuous, because the function is undefined at $x = 0$ .

2. For the piecewise function $f(x) = \begin{cases} x^2, & x \leq 1 \\ 2x - 1, & x > 1 \end{cases}$ , determine continuity at $x = 1$ and the type of discontinuity (if any)

Solution:

$\lim\limits_{x \to 1^-} f(x) = 1^2 = 1$
$\lim\limits_{x \to 1^+} f(x) = 2 \times 1 - 1 = 1$
$f(1) = 1$ All three are equal, so the function is continuous at $x=1$ .

Answer: Continuous at $x=1$ , no discontinuity.

3. Let $f(x)$ be continuous on $[0, 2]$ with $f(0) = -1, f(2) = 3$ . Prove that there exists $x_0$ in $(0, 2)$ such that $f(x_0) = 0$

Solution: By the zero point theorem, $f(0) \cdot f(2) = -1 \times 3 < 0$ , so there exists $x_0 \in (0, 2)$ such that $f(x_0) = 0$ .

Answer: There exists $x_0 \in (0, 2)$ such that $f(x_0) = 0$ .

4. Is $f(x) = \sin x$ continuous on $\mathbb{R}$ ?

Solution: $\sin x$ is an elementary function and is continuous everywhere on $\mathbb{R}$ .

Answer: Continuous everywhere on $\mathbb{R}$ .

5. If $f(x)$ is continuous on $[a, b]$ , is it necessarily bounded and does it attain its maximum and minimum values?

Solution: By the boundedness and extreme value theorems, a continuous function on a closed interval is always bounded and attains its maximum and minimum values.

Answer: Always bounded, and attains maximum and minimum values.

Continuity

Continuity

Continuity of Functions

Types of Discontinuities

Continuity of Elementary Functions

Properties of Continuous Functions on Closed Intervals

Exercises

1. Is f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ continuous at x=0x = 0x=0? Explain why.

2. For the piecewise function f(x)={x2,x≤12x−1,x>1f(x) = \begin{cases} x^2, & x \leq 1 \\ 2x - 1, & x > 1 \end{cases}f(x)={x2,2x−1,​x≤1x>1​, determine continuity at x=1x = 1x=1 and the type of discontinuity (if any)

3. Let f(x)f(x)f(x) be continuous on [0,2][0, 2][0,2] with f(0)=−1,f(2)=3f(0) = -1, f(2) = 3f(0)=−1,f(2)=3. Prove that there exists x0x_0x0​ in (0,2)(0, 2)(0,2) such that f(x0)=0f(x_0) = 0f(x0​)=0

4. Is f(x)=sin⁡xf(x) = \sin xf(x)=sinx continuous on R\mathbb{R}R?

5. If f(x)f(x)f(x) is continuous on [a,b][a, b][a,b], is it necessarily bounded and does it attain its maximum and minimum values?

1. Is $f(x) = \frac{1}{x}$ continuous at $x = 0$ ? Explain why.

2. For the piecewise function $f(x) = \begin{cases} x^2, & x \leq 1 \\ 2x - 1, & x > 1 \end{cases}$ , determine continuity at $x = 1$ and the type of discontinuity (if any)

3. Let $f(x)$ be continuous on $[0, 2]$ with $f(0) = -1, f(2) = 3$ . Prove that there exists $x_0$ in $(0, 2)$ such that $f(x_0) = 0$

4. Is $f(x) = \sin x$ continuous on $\mathbb{R}$ ?

5. If $f(x)$ is continuous on $[a, b]$ , is it necessarily bounded and does it attain its maximum and minimum values?