Question

Is it possible for a function to be continuous at all points in its domain and also have a one-sided limit equal to +infinite at some point?

Answer 1

Yes, it is possible. (But the point at which the limit is infinite cannot be in the domain of the function.)

Explanation:

Recall that `f` is continuous at `a` if and only if `lim_(xrarra)f(x) = f(a)`.

This requires three things:

1) `lim_(xrarra)f(x)` exists.
Note that this implies that the limit is finite. (Saying that a limit is infinite is a way of explaining why the limit does not exist.)

2) `f(a)` exists (this also implies that #f(a) is finite).

3) items 1 and 2 are the same.

Relating to item 1 recall that `lim_(xrarra)` exists and equals `L` if and only if both one-sided limits at `a` exist and are equal to `L`

So, if the function is to be continuous on its domain, then all of its limits as `xrarra^+` for `a` in the domain must be finite.

We can make one of the limits `oo` by making the domain have an exclusion.

Once you see one example, it's fairly straightforward to find others.

`f(x) = 1/x`

Is continuous on its domain, but `lim_(xrarr0^+)1/x = oo`