Name: Functions continuous on all real numbers | Limits and continuity | AP Calculus AB | Khan Academy
Uploaded: 2022-07-05T13:44:33.000Z
Duration: 3 min 17 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

- [Voiceover] Which of the following functions

Present continuous

A function is going to be continuous over some interval.

doesn't have any jumps or discontinuities

and it for sure has to be defined over that interval

a continuous function could look something like this.

let me make that line a little bit thicker,

so this function right over here is continuous.

over an interval, or non-continuous functions,

They could have some type of an asymptotic discontinuity

so they could have a gap where they're not defined,

or maybe they actually are defined there,

so all of these are examples of discontinuous functions.

Now, if you want the more mathy understanding of that

is equal to the value of the function at a,

so once again, in order to be continuous 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,

so let's think about f of x equals e to the x.

most of the common functions that you've learned in math,

they don't have these strange jumps or gaps

but things like e to the x, it doesn't have any of those.

it's really more of getting this intuitive sense

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

so it's reasonable to say that it's continuous

but you could do a more rigorous proof if you like