# Give a small proof of the above?

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If the range of a function #f# is #RR# and #f# is strictly monotone in #RR# then #f# is continuous in #RR#

If the range of a function

##### 1 Answer

I'm going to assume that by "range" you are referring to a function's **image** and not the function's **codomain**.

A strictly monotonic function is either strictly non-decreasing or strictly non-increasing. We consider whether such a function can have discontinuities.

A monotonic function cannot have removable discontinuities because having one would imply that at the particular value

A monotonic function cannot have an essential discontinuity because, if it did, then there would be some

A monotonic function can have jump discontinuities because such continuities do not necessarily cause a change in the sign of

These are the only types of discontinuities and our given constraints do not allow any of them to occur. Thus, our function is continuous.