# 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?

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 ${\lim}_{x \rightarrow a} f \left(x\right) = f \left(a\right)$.

This requires three things:

1) ${\lim}_{x \rightarrow a} f \left(x\right)$ 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) $f \left(a\right)$ exists (this also implies that #f(a) is finite).

3) items 1 and 2 are the same.

Relating to item 1 recall that ${\lim}_{x \rightarrow a}$ 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 $x \rightarrow {a}^{+}$ for $a$ in the domain must be finite.

We can make one of the limits $\infty$ by making the domain have an exclusion.

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

$f \left(x\right) = \frac{1}{x}$

Is continuous on its domain, but ${\lim}_{x \rightarrow {0}^{+}} \frac{1}{x} = \infty$