L'Hôpital's Rule/Also defined as

L'Hôpital's Rule: Also defined as

In many elementary texts on real analysis, it appears to be commonplace for $f$ and $g$ to be specified as being continuous on $\closedint a b$.

However, this is not strictly necessary, as differentiability on $\openint a b$ is completely adequate.

If is of course noted that from Differentiable Function is Continuous, if $f$ and $g$ are differentiable on $\openint a b$, they are also continuous on $\openint a b$, just not necessarily also at $a$ or $b$.

Indeed, for L'Hôpital's Rule: Corollary $2$, $f$ and $g$ are demonstrably not continuous either at $a$ or $b$, or possibly both.

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 11.8 \ (3)$