Functions and Continuity

Functions and Continuity Overview

1. Introduction to Function Continuity

• Defining Continuity

  • A function $y = f(x)$ is said to be continuous at the point $x_0$ in its neighborhood if:

  • 
    ext{lim}{x o x0} f(x) = f(x_0)

  • Here, $x_0$ is referred to as a continuous point of $f(x)$.

2. Operations of Continuous Functions

• Continuous Function Properties

  • If $f(x)$ and $g(x)$ are continuous at $x0$, then their sum, difference, product, and quotient (when the denominator is not zero) are also continuous at $x0.

3. Points of Discontinuity

• Types of Discontinuities

  • Identified through three scenarios:

  • (1) The function $f(x)$ is not defined at $x_0.

  • (2) The function $f(x)$ has a defined value at $x0$, but the limit as $x$ approaches $x0$ does not exist.

  • (3) The limit at $x0$ exists, but it does not equal $f(x0).

4. Characteristics of Continuous Functions on Closed Intervals

• Properties

  • Extreme Value Theorem: A continuous function on a closed interval $[a, b]$ will attain both a maximum and minimum value.

  • Example: If $f(x)$ is continuous on $[a,b]$, there exist $m$ and $M$ such that:
    f(x{min}) ext{ and } f(x{max})

• Significance: Closed intervals ensure that functions are bounded.

5. Continuous Function Operations

• Operations on Continuous Functions

  • Theorem 2 states if $f(x)$ and $g(x)$ are continuous at $x_0$, then:

  • $f(x) + g(x)$, $f(x) - g(x)$, $f(x)g(x)$, and $ rac{f(x)}{g(x)}$ (if $g(x)
    eq 0$) are also continuous at $x_0.$

6. Left and Right Continuity

• Definitions

  • A function $y = f(x)$ is left-continuous at $x_0$ if:

  • ext{lim}{x o x0^-} f(x) = f(x_0)

  • A function is right-continuous at $x_0$ if:

  • ext{lim}{x o x0^+} f(x) = f(x_0)

7. Examples of Continuity

Example 1

• Prove that the function $f(x) = egin{cases} x ext{sin} rac{1}{x}, & x
  eq 0 \ 0, & x = 0 \ ext{is continuous at } x = 0 \ ext{lim }_{x o 0} f(x) = 0 = f(0)

Example 2

• Prove function $f(x) = x^2$ is continuous at $x = 2$:

  • Method 1 :

    • By definition, $f(x)$ is continuous.

  • Method 2 :

    • Showing:

      • ext{lim}_{ riangle x o 0} (4 riangle x + ( riangle x)^2)= 0$$