Question related to Darboux's theorem Darboux's theorem says that $f'$ has intermediate value property. More precisely,

( Darboux's theorem) If $f$ is differentiable on $[a,b]$, and if r is any number for which $f'(a)<r<f'(b)$ then $\exists$ c in (a,b) such that $f'(r)=c$.

Thus Darboux's theorem implies that $f'$ cannot have any simple discontinuities on $[a,b].$
I have a question as follows:

Is there a function $f$ satisfying $\displaystyle \lim_{x\to d} f'(x)=\infty$ for some $d \in (a,b)$ under the assumptions in Darboux's theorem?

If so, I can conclude that for each $x \in (a,b),$ either $(i)$ $f'$ is continuous at $x$ or $(ii)$ $f'$ oscillates near $x.$
It seems that there is no differentiable function $f$ on $[a,b]$ satisfying function $\displaystyle \lim_{x\to d} f'(x)=\infty$ and $\exists f'(d)$ for some $d \in (a,b)$.
Please let me know if you have any idea or comment for my question. Thanks in advance!

No, there can't be.
If $f'(x)\to +\infty$ for $x\to d$, then there will be $\delta$ such that $f'(x) > f'(d)+2$ on $(d, d+\delta)$. And this contradicts Darboux's theorem on $[d,d+\delta/2]$ -- there won't be any place with $f'(x) = f'(d)+1$.
Be a bit careful about the precise meaning of "oscillates" in your conclusion (ii), though. It is possible for $f'$ to have a strict global minimum at $d$, for example.
