Here we use limits to ensure piecewise functions are continuous.

- Constant function
- Polynomials
- Rational functions
- Power function
- Exponential function
- Logarithmic function
- Trig functions
- Inverse trig functions

We proved continuity of polynomials earlier using the Sum Law, Product Law and
continuity of power functions.

We proved continuity of rational functions earlier using the Quotient Law and
continuity of polynomials.

We can prove continuity of the remaining four trig functions using the Quotient Law
and continuity of sine and cosine functions.

Since a continuous function and its inverse have ”unbroken” graphs, it follows that an
inverse of a continuous function is continuous on its domain.

This implies that inverse trig functions are continuous on their domains.

In this section we will work a couple of examples involving limits, continuity and piecewise functions.

Consider the next, more challenging example.

Looking at the limit from the right, we have

Hence for this function to be continuous at , we must have that

Hmmmm. More work needs to be done.

To find and that make is continuous at , we need to find and such that Looking at the limit from the left, we have

Looking at the limit from the right, we have

Hence for this function to be continuous at , we must have that

So now we have two equations and two unknowns: Set and write

hence Let’s check, so now plugging in values for both and we find Now and So setting and makes continuous at and . We can confirm our results by looking at the graph of :