Continuity Help

Introduction to Continuity

Let f be a function whose domain contains the interval (a, b). Assume that c is a point of (a, b). We say that the function f is continuous at c if

Conceptually, f is continuous at c if the expected value of f at c equals the actual value of f at c .

Examples

You Try It: Discuss continuity of the function

We note that Theorem 2.1 guarantees that the collection of continuous functions is closed under addition, subtraction, multiplication, division (as long as we do not divide by 0), and scalar multiplication.

Math Note: If f ο g makes sense, if lim _{x → c} g ( x ) = ℓ , and if lim _{s → ℓ} f ( s ) = m , then it does not necessarily follow that lim _{x → c} f ο g ( x ) = m . [We invite the reader to find an example.] One must assume, in addition, that f is continuous at ℓ. This point will come up from time to time in our later studies.

In the next section we will learn the concept of the derivative. It will turn out that a function that possesses the derivative is also continuous.

Find practice problems and solutions for these concepts at: Foundations of Calculus Practice Test.