I'm constructing a continuous function $f$ which is well defined on $\mathbb R$ and its derivative $g$ does not exists on a countable infinite points. This kind of functions is easy to find. For example, a continuous function consisting a series of connected line segments. I think the $g$ is actually continuous on its domain, if its domain is restricted on where it exists (correct me if I am wrong). Here its domain is a dense set. Then I am wondering if we can find an example or a counterexample which succeed or fail the L'Hospital rule, of which the numerator and denominator are continuous on $\mathbb R$, and their derivative exists almost everywhere.
Update:
I guess I should be more specific. I want the points where the derivative does not exist are concentrated around the point where L' Hospital will be used. For example, L' Hospital rule will be used at $x=0$ and the numerator and denominator's derivatives do not exist on $\{x=\frac{1}{n}\}$
Update2:
I note that in common L'Hospital rule statement and proof, it is assumed that in $\lim\frac{f(x)}{g(x)}$, $f$ and $g$ are not only continuous on an interval (neighborhood of $x_0$, where the rule will be used), but also differentiable on the interval everywhere. See here. I am wondering if this condition is too strong. I think the derivatives of $f$ and $g$ are not necessary to exist everywhere on the interval but only need the $\lim \frac{f'}{g'}$ to exist, to apply L'Hospital rule. In other words, can we relax the condition by allowing $f'$ and $g'$ to exist outside a smaller, in some sense (e.g. measure zero ?), subset of the interval.