If a real function $f:ℝ→ℝ$ is twice differentiable at a point $x$, then the first derivative must be continuous at $x$, and assuming $f′(x)>0$, then there exist $δ>0$ such that $f′(y)>0 $ for all $y∈(x−δ,x+δ)$, then on this interval $f$ must be increasing. Repeating this process for all $x$, we conclude that $δ$ is a function of $x$


Assuming that this function is analytic (the stronger form). I am asking if $δ=δ(x)$ is a continuous function in $x$.


If no, then what are the conditions on the function $f$ such that the above property holds true.



Is $δ=δ(x)$ a continuous function
$\delta(x)$ (defined as the maximum of appropriate $\delta$'s) is even 1-Lip (for any $f$). Indeed, if $|x-y|=a$, then $\delta(y)\geqslant \delta(x)-a$, since $(y-c,y+c)\subset (x-\delta(x),x+\delta(x))$ for $c=\max(\delta(x)-a,0)$. Analogously $\delta(x)\geqslant \delta(y)-a$ and therefore $|\delta(x)-\delta(y)|\leqslant a$. 


