Question

What does it mean for a function to be continuous on an interval?

Short answer

Answer: A function is continuous on an interval if its limit as x approaches any point in the interval exists and equals the function's value at that point, meaning there are no gaps or jumps in the function. Some properties of continuous functions include the sum, difference, product, and quotient of two continuous functions being continuous, their graphs appearing unbroken, and their compliance with the Intermediate Value Theorem. Examples of continuous functions are polynomial functions, trigonometric functions, and exponential and logarithmic functions.

Step-by-step solution

Definition of Continuity

A function f(x) is said to be continuous on an interval [a, b] if for all x in [a, b], the limit of f(x) as x approaches c exists and is equal to f(c) for all c in [a, b]. In mathematical terms, this is expressed as: Limit as x approaches c of f(x) = f(c)

Properties of Continuous Functions

Continuous functions have several properties, such as: 1. The sum, difference, product and quotient of two continuous functions are continuous. 2. Continuous functions are unbroken and have no gaps or jumps in their graphs. 3. The Intermediate Value Theorem applies to continuous functions, which states that if f(x) is continuous on the interval [a, b], and there exists a number k between f(a) and f(b), then there exists a number c in (a, b) such that f(c) = k.

Examples of Continuous Functions

Some examples of continuous functions are: 1. Polynomial functions: These functions have no gaps or jumps and are continuous over their entire domain. 2. Trigonometric functions: Functions like sine and cosine are continuous over their entire domain. 3. Exponential and logarithmic functions: These functions are continuous over their entire domain. Overall, a function is considered continuous on an interval if it does not have any gaps or jumps, and its limit as x approaches any point in the interval exists and equals the function's value at that point.