Continuous Functions
To get an intuitive feel for what a continuous function is, you can think of a continuous function as one for which you can draw the graph without picking up your pencil. It has no holes or jumps in the graph. This is a good picture to have in your head; however, the official definition of a continuous function is slightly more involved.
A function f(x) is continuous at a point c if lim x -> c f(x) = f(c). A function f is continuous if it is continuous at every point c in its domain.
For some functions, you will be able to say they are continuous right away. All polynomials are continuous. The sine and cosine functions are continuous. The tangent function however is not continuous.
Many times you will be asked to show that a piecewise-defined function is continuous. Typically, each piece of the function is continuous, so we don't need to worry about most of the domain. However, there are the "suspicious points". These include the points c at which the overall function changes from being defined by piece A to being defined by piece B.
To determine whether the function is continuous, we make a list of all suspicious points, c. Then we look at lim x -> c f(x) and f(c). For f to be continuous at c, these two quantities need to be the same.
Many times, f(x) will be defined by a different formula when x > c, versus when x < c. In these cases, you need to compare lim x ->c- f(x), f(c), and
lim x ->c+ f(x). Again, for the function to be continuous, all these quanities need to be the same.
The reason for needing to check the right-hand limit as well as the left-hand limit is the rule that:
lim x -> cf(x) = L if and only if lim x ->c-f(x) = L = lim x ->c+ f(x). This is an important fact to remember.