# How to Determine Whether a Function Is Continuous

A graph for a function that's smooth without any holes, jumps, or asymptotes is called *continuous.* Your pre-calculus teacher will tell you that three things have to be true for a function to be continuous at some value *c* in its domain:

The function must exist at an*f*(*c*) must be defined.*x*value (*c*), which means you can't have a hole in the function (such as a 0 in the denominator).**The limit of the function as**The left and right limits must be the same; in other words, the function can't jump or have an asymptote. The mathematical way to say this is that*x*approaches the value*c*must exist.must exist.

**The function's value at***c*and the limit as*x*approaches*c*must be the same.

For example, you can show that the function

is continuous at *x *= 4 because of the following facts:

You can substitute 4 into this function to get an answer: 8.*f*(4) exists.If you look at the function algebraically, it factors to this:

Nothing cancels, but you can still plug in 4 to get

which is 8.

Both sides of the equation are 8, so 'f(x) is continuous at x = 4.

If any of the above situations aren't true, the function is discontinuous at that value for *x*.