Additional Resources
OpenStax Online Textbook
How to Read a Math Textbook
Additional Instructional Resources
Objectives
- Explain the three conditions for continuity at a point.
- Describe three kinds of discontinuities.
- Define continuity on an interval.
- State the theorem for limits of composite functions.
- Provide an example of the intermediate value theorem.
Summary
A function has limit as if and only if has a left-hand limit at has a right-hand limit at and the left- and right-hand limits are equal. Visually, this means that there can be a hole in the graph at but the function must approach the same single value from either side of
A function is continuous at whenever is defined, has a limit as and the value of the limit and the value of the function agree. This guarantees that there is not a hole or jump in the graph of at So we say that 𝑓 is continuous at point 𝑥=𝑎 whenever
lim𝑥→𝑎 𝑓(𝑥) = 𝑓(𝑎)
that is to say that the following conditions are met:
1. lim𝑥→𝑎− 𝑓(𝑥) = 𝑓(𝑎) and
2. lim𝑥→𝑎+ 𝑓(𝑥) = 𝑓(𝑎).
A function is differentiable at whenever exists, which means that has a tangent line at and thus is locally linear at Informally, this means that the function looks like a line when viewed up close at and that there is not a corner point or cusp at
Of the three conditions discussed in this section (having a limit at being continuous at and being differentiable at the strongest condition is being differentiable, and the next strongest is being continuous. In particular, if is differentiable at then is also continuous at and if is continuous at then has a limit at
See the Desmos demonstration about Continuity.
See the Desmos demonstration about Continuity of Piecewise Functions.