Here we use limits to check whether piecewise functions are continuous.
- Constant functions
- Rational functions
- Power functions
- Exponential functions
- Logarithmic functions
- Trigonometric functions
- Inverse trigonometric functions
In essence, we are saying that the functions listed above are continuous wherever they are defined.
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.
Using the Limit Laws we can prove that given two functions, both continuous on the same interval, then their sum, difference, product, and quotient (where defined) are also continuous on the same interval (where defined).
Therefore, which proves that is continuous at .
To find and that make is continuous at , we need to find and such that Since , it follows that Looking at the limit from the left, we haveLooking at the limit from the right, we have Hence 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 :