Calculus > Check the Continuity of a Function

A continuous function has a graph without gaps, splits, or holes. In other words, you could draw a line of the graph on paper without lifting the pencil. You sometimes need to check for continuity of a function in order to perform some operations in calculus, such as applying Rolle’s Theorem. Polynomial functions, rational functions, power functions, the trigonometric functions sin(x) and cos(x), logarithmic functions, the function e^{x} and exponential functions are all continuous functions over the **entire domain**. There are a few general rules you can refer to when trying to determine if your function is continuous. For other functions, you need to do a little detective work.

## How to check for the continuity of a function

Step 1: **Draw the graph with a pencil** to check for the continuity of a function. If your pencil stays on the paper from the left to right of the entire graph, without lifting the pencil, your function is continuous. In other words, if your graph has gaps, holes or is a split graph, your graph isn’t continuous. If you aren’t sure about what a graph looks like if it’s not continuous, check out the images in this article: When is a Function Not Differentiable?

Step 2: **Figure out if your function is one of the following types:** a power function, exponential function, logarithmic function, polynomial function, rational function (the ratio of two polynomial functions), the function e^{x} or a trigonometric function sin(x) or cos(x). If it is, then there’s no need to go further; your function is continuous.

Step 3: **Check if your function is the sum difference, or product** of a continuous function listed in Step 2. If it is, your function is continuous. For example, sin(x)*cos(x) is the product of two continuous functions and so is continuous.

Step 4: **Check quotient functions for the possibility of zero as a denominator**. The quotient f(x)/g(x) of continuous functions is continuous at all points x where the denominator isn’t zero.

*That’s it!*

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