The **continuity test is a set of three conditions** that tell you whether a function is continuous at a specific point.

A function is continuous at an x-value of “c” if all of the following conditions are true:

**The function is defined at c**: In other words, if you put the “x” value into the function, you’ll get a real value and not, for example, division by zero or something else that’s undefined.**The function approaches the same value from the left and the right**. In other words, the limit from above and the limit from below are in agreement.**The function approaches the value f(c) from left and right.**This is slightly different from #2, which just asks if they approach the same value. This condition states that the “same value” in condition 2 is the value you get when you plug the x-value into the function. In other words, conditions 1 and 2 should equal the same y-value.

## Continuity Test Examples

**Example question:** Is the function f(x) = 3x^{2} + 7 continuous at x = 1?

**Solution**: Check the three conditions given in the definition.

**Is the function defined at c?**Replace “x” in the formula to give:

f(x) = 3(1^{2}) + 7 = 10.*Yes, the function is defined at x = 1*.**Does the function approach the same value from the left and the right?**. There are a couple of ways to check this:- Graph the function and check to see if both sides approach the same number.

This graph shows that both sides approach f(x) = 16, so the function meets this part of the continuity test. - You can also create a table of values for small increments close to x = 1:

The numbers in the table are getting close to 10 from both sides (from the negative and positive directions).

- Graph the function and check to see if both sides approach the same number.
**Does the function approach the value f(c) from above and below?**In this step, compare the values from Step 1 and Step 2 above. Yes, the two sides meet at y = 10.

This function has passed the continuity test. We can conclude that the function is continuous.