I understand the conditions for Rolle's theorem in english : a function $f$ has to be continous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$. But I'm studying in Russian, and they write the conditions like this:
a) function $f$ is continous on $[a,b]$,
b) function $f$ has at all points of the interval $(a,b)$ a finite or definite sign infinite derivative,
I'm having trouble understanding point b), what does 'definite sign infinite derivative' mean? Is there a difference between that, and simply being differentiable on $(a,b)$? Maybe the translation isn't quite right, my russian isn't perfect, but I think that's what it means in English.
All I could think of was, maybe they are refering to when the limit of $f(x)$ when $x \to a$ or $x \to b$ is equal to $\pm\infty$, but then $f$ wouldn't be continous on $[a,b]$. So, I'm lost.
Do you guys have any ideas? Thanks a lot in advance.