are continuous for all real numbers?

So let's just remind ourselves

what it means to be continuous.

What a continuous function looks like.

So, a continuous function,

let's see, that's my y-axis,

that is my x-axis.

A function is going to be continuous over some interval.

If it just has,

doesn't have any jumps or discontinuities

over that, or gaps over that interval,

so if it's connected

and it for sure has to be defined over that interval

without any gaps,

so 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,

that makes it discontinuous.

They could have

a jump of 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 removable discontinuity,

so all of these are examples of discontinuous functions.

Now, if you want the more mathy understanding of that

and we've looked at this before,

we say that a function f is continuous,

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, 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 over 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,

e to the x looks something like this,

it's defined for all real numbers,

there's no jumps or gaps of any kind

and so,

this f of x is continuous

for all real numbers

and f only.

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 of getting this intuitive sense

of like, look, e to the x is defined for all real numbers

and so,

and 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.