Continuity

Continuity at a Point

Continuity is a fundamental concept in calculus that describes whether a function behaves predictably at a specific point. A function f(x) is said to be continuous at a point x = c if it satisfies three essential conditions:

• f(c) exists: The function is defined at c (i.e., c is in the domain of f).

• \lim_{x \to c} f(x) exists:

• \lim_{x \to c} f(x) = f(c): The limit equals the function value at c.

If any of these conditions fail, the function f is discontinuous at c, and a discontinuity occurs.

Types of Discontinuities

Discontinuities can occur in several ways, such as:

• Jump Discontinuity: The left and right limits exist but are not equal.

• Infinite Discontinuity: The function approaches infinity at the point.

• Removable Discontinuity: The limit exists, but the function value does not match the limit.

Right and Left Continuity

Sometimes, a function may be continuous only from one side at a point. The definitions are:

• Right-continuous at c:

• Left-continuous at c:

Continuity of Common Functions

Functions Continuous Everywhere in Their Domains

The following types of functions are continuous at every number in their domains:

• Polynomials

• Rational functions (except where the denominator is zero)

• Radical/root functions (where the radicand is defined)

• Trigonometric functions

• Inverse trigonometric functions

• Logarithmic functions (where the argument is positive)

• Exponential functions

• Compositions of the above functions (in their common domains)

Examples and Applications

Example 1: Identifying Points of Discontinuity

Given a graph or formula, determine where f is not continuous by checking the three continuity conditions.

Example 2: One-Sided Continuity

For each discontinuity, determine if f is continuous from the right, left, or neither by evaluating the one-sided limits.

Example 3: Continuity of a Composite Function

Consider . The function is continuous wherever the denominator is not zero and the numerator is defined. Thus, f is continuous for , , .

Example 4: Continuity Test at a Specific Point

For at :

• exists:

• exists: The function is defined and continuous at .

• : True.

Thus, f is continuous at .

Example 5: Piecewise Function Continuity

For at :

Since the left and right limits are not equal, f is not continuous at .

Example 6: Interval of Continuity

For , find where f is continuous. The function is continuous everywhere except possibly at . Check the limit and function value at to determine continuity.

Example 7: Continuity of a Piecewise Function with a Constant

For , find values of for which f is continuous everywhere. Set the left and right limits at equal and solve for :

• Set and solve for .

Additional info: These examples illustrate the application of the continuity test and the importance of checking both the function value and the limit at a point, especially for piecewise functions.