I'd like to know if $\exists$ $\displaystyle \lim_{t\to 0^+}u(t) u'(t)$ for a function $u \in C^1(0,1)\cap C[0,1]$ with $u(0)=0$. If $u \in C^1[0,1),$ it's trivial. I'm interested in the case $u \not \in C^1[0,1).$ For example, if $u(t)=t^{\frac{1}{2}},$ then $u'(t)=\frac{1}{2}t^{-\frac{1}{2}}.$ Consequently $u(t) u'(t)=\frac{1}{2}$ for $t \in (0,\infty),$ and $\displaystyle\lim_{t\to 0^+}u(t) u'(t)=\frac{1}{2}.$
Are there any general result for this question? [No general result : see below user296602 comment]
If there is no result for general functions, we may think
(1) $u$ is a concave function on $[0,1].$ [It's false: see below G.Sassatelli comment]
Or
(2)$u=u(t)$ is a solution to a singular differential equation, e,g., $u$ is as solution to problem $u''+h(t)f(u)=0,~t \in (0,1)$ with $u(0)=u(1)=0$, where $h \not \in L^1(0,1)$ and $|f(u)|\le C|u|$ for $u \in \mathbb{R}.$
Please let me know if you have any idea or comment for this question.
Thanks in advance!