Continuity and Types of Discontinuities in Functions

Continuity of Functions

Definition of Continuity

A function f is said to be continuous at a point x = a if the following condition holds:

• Definition: $\lim_{x \to a} f(x) = f(a)$

This means that the function's value at a matches the value approached by f(x) as x gets arbitrarily close to a.

Note: A function is continuous on its domain if it is continuous at every point in that domain.

Types of Discontinuities

Discontinuities occur when a function is not continuous at a point. The main types are:

• Jump Discontinuity: The function 'jumps' from one value to another at a certain point.

• Removable Discontinuity: There is a 'hole' in the graph, often because the function is not defined at that point, but the limit exists.

• Vertical Asymptote: The function approaches infinity or negative infinity as x approaches a certain value.

Intervals of Continuity

Determining Intervals of Continuity

To find where a function is continuous, identify points of discontinuity and exclude them from the domain.

• Example: For a function with jump discontinuities at x = -7 and x = -5, the intervals of continuity are: $(-\infty, -7) \cup (-7, -5) \cup (-5, \infty)$

• Union Notation: The union symbol $\cup$ is used to combine intervals where the function is continuous.

Vertical Asymptotes and Rational Functions

Vertical asymptotes occur in rational functions where the denominator is zero.

• Example: For $f(x) = \frac{x+8}{(x-8)(x-2)}$, vertical asymptotes at x = 8 and x = 2 are points of discontinuity.

• Intervals of Continuity: $(-\infty, 2) \cup (2, 8) \cup (8, \infty)$

No Real Discontinuities

If the denominator of a rational function does not have real roots, there are no vertical asymptotes, and the function is continuous everywhere on the real line.

• Example: $x^2 + 36 = 0$ has no real solutions, so the function is continuous for all real numbers: $(-\infty, \infty)$

Continuity of Composite and Piecewise Functions

Domain and Continuity of Composite Functions

For composite functions, determine the domain by considering the domains of both the inner and outer functions.

• Example: $f(x) = \sqrt{x-2}$ is defined for $x \geq 2$; $g(x) = \sqrt{x+5}$ is defined for $x \geq -5$.

• Domain of $f$: $[2, \infty)$

• Domain of $g$: $[-5, \infty)$

Checking Continuity for Piecewise Functions

To check continuity at a partition point in a piecewise function, compare the left and right limits to the function value at that point.

• If $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$, the function is continuous at a.

• If the limits from the left and right do not match, or do not equal $f(a)$, there is a discontinuity.

Example: Piecewise Function Discontinuity

• For $f(x)$ defined piecewise, if $\lim_{x \to -1^-} f(x) = 3$ and $\lim_{x \to -1^+} f(x) = 7$, then $f(x)$ is discontinuous at $x = -1$ because the two limits are not equal.

Summary Table: Types of Discontinuities

Type	Description	Example
Jump	Function jumps to a different value	Piecewise function with different values at partition
Removable	Hole in the graph, limit exists but function not defined	$f(x) = \frac{x^2-1}{x-1}$ at $x=1$
Vertical Asymptote	Function approaches infinity	$f(x) = \frac{1}{x-2}$ at $x=2$

Key Formulas

• Continuity at a Point: $\lim_{x \to a} f(x) = f(a)$

• Domain of Square Root Function: $x \geq$ value inside the root

• Intervals of Continuity: Exclude points of discontinuity from the domain

Additional info:

• Union notation $\cup$ is used to combine intervals of continuity.

• Piecewise functions require checking both left and right limits at partition points.