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:

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:

• Union Notation: The union symbol 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 , vertical asymptotes at x = 8 and x = 2 are points of discontinuity.

• Intervals of Continuity:

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: has no real solutions, so the function is continuous for all real numbers:

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: is defined for ; is defined for .

• Domain of :

• Domain of :

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 , the function is continuous at a.

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

Example: Piecewise Function Discontinuity

• For defined piecewise, if and , then is discontinuous at because the two limits are not equal.

Summary Table: Types of Discontinuities

Key Formulas

• Continuity at a Point:

• Domain of Square Root Function: value inside the root

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

Additional info:

• Union notation is used to combine intervals of continuity.

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