Functions continuous on all real numbers | Limits and continuity | AP Calculus AB | Khan Academy

Which of the following functions are continuous for all real numbers? So let's just remind ourselves what it means to be continuous, what a continuous function looks like.

A continuous function—let's say that's my Y-axis, that is my X-axis—a function is going to be continuous over some interval if it just doesn't have any, uh, jumps or discontinuities over that or gaps over that interval. So, if it's connected, it for sure has to be defined over that interval without any gaps.

For example, a continuous function could look something like this. This function—let me make that line a little bit thicker—so this function right over here is continuous. It is connected over this interval, the interval that we can see now.

Examples of discontinuous functions over an interval, or non-continuous functions, well, they would have gaps of some kind. They could have some type of an asymptotic discontinuity, so something like that makes it discontinuous. They could have jump discontinuity, something like that. They could just have a gap where they're not defined, so they could have a gap where they're not defined, or maybe they actually are defined there, but it's a removable discontinuity.

So all of these are examples of discontinuous functions. Now, if you want the more mathematical understanding of that—and we've looked at this before—we say that a function f is continuous at some value x equals a if and only if, draw my little two-way arrows here, say if and only if the limit of f of x as x approaches a is equal to the value of the function at a.

So once again, in order to be continuous there, you at least have to be defined there. Now, when you look at these, the one thing that jumps out at me is that in order to be continuous for all real numbers, you have to be defined for all real numbers. And g of x is not defined for all real numbers; it's not defined for negative values of x, and so we would rule this one out.

So let's think about f of x equals e to the x. It is defined for all real numbers, and as we'll see, most of the common functions that you've learned in math, they don't have these strange jumps or gaps or discontinuities. Some of them do—functions like 1/x and things like that—but things like e to the x, it doesn't have any of those.

We could graph e to the x; e to the x looks something like this. It's defined for all real numbers; there's no jumps or gaps of any kind. So this f of x is continuous for all real numbers.

Now, I didn't do a very rigorous proof. You could if you like, but for the sake of this exercise, it's really more about getting this intuitive sense of, like, look, e to the x is defined for all real numbers, and so there's no jumps or gaps here. So it's reasonable to say that it's continuous. But you could do a more rigorous proof if you like as well.