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?

1 Answer
Jim H
Jul 27, 2015

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#

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