# Continuity of Functions

### Example 1

The function *f*(*x*) = 4*x* + 1 is a line, therefore it's continuous everywhere. In particular, it's continuous at *x* = 5 with *f*(5) = 21. How must the *x*-values be restricted if we want to have 20.5< *f*(*x*) < 21.5?

### Example 2

The function

is discontinuous at *x* = 1. Explain how this function fails the continuous function guarantee.

### Example 3

For the given function, value of *c*, and specified range of *f*(*x*), find an appropriate restriction of *x* that produces the kind of picture we want.

Remember that there are multiple right answers for each of these. As long as our answer produces the correct sort of graph, it's fine.