Module 2 Content Sheet Part 1

Math 180 Worksheets

W3

Keywords: continuity, graphing, intermediate value theorem, limit definition of the deriva-

tive, secant lines

3 Module 2 Content Sheet Part 1

2.6 Continuity

3.1 Introducing the Derivative

3.2 Working with Derivatives

�Watch the Panopto Video on Introduction to Continuous Functions

2.6 Continuity In words, a function fis continuous at a number aif the graph can be

sketched without lifting a pencil at and near the point a.

Question 1 Sketch two graphs, one that is continuous everywhere, another that is not con-

tinuous at a point.

Definition: A function fis continuous at a number aif lim

x!af(x)=f(a).

Compare the definition with your two graphs above, does the continuous function satisfy

lim

x!af(x)=f(a)foralla? and what condition does your example of a non continuous

function not satisfy?

Most familiar functions are continuous on their domain. Every polynomial function is

continuous, as are the sine, cosine, and exponential functions. The restriction for rational

functions, tangent, secant, logarithms are that they are only continuous on their domain.

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Math 180 Worksheets

W3

For example: f(x)= 2

x3is a rational function that is continuous at all points x6=3.

At x= 3 the function is undefined undefined, and thus cannot be continuous.