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Functions continuous on all real numbers

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.B (LO)
,
LIM‑2.B.1 (EK)
,
LIM‑2.B.2 (EK)

Video transcript

which of the following functions are a 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 say that's my y-axis that is my x-axis a function is going to be continuous over some interval if it just says it doesn't have any jumps or discontinuities over that or gaps over that interval so if it's connected or and it for sure has to be defined over that interval without any gaps so for example an example 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 discontinuity something like that they could just have a gap where they're where they're not defined so they could have a gap where they're not defined or maybe it's 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 the 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 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 the strange jumps or gaps or discontinuities some of them do functions like 1 over X and and things like that but things like e to the X it doesn't have any of those we could graph eetu 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 e 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 so like look it look either the X is defined for all real numbers and so and there's no jumps or gaps here so it's reasonable say that it's continuous but you could do a more rigorous proof if you like as well