Relationship between semiconvexity and Lipschitz continuity

Question

The following is written in the book I read:

Let $T>0$ and $f:[0,T]\to\mathbb{R}$ be a semiconvex function in $[0,T]$. Then $f$ is Lipschitz continuous in $[0,T]$.

I know the proof that $f$ is locally Lipschit continuous in $(0,T)$. I however have no idea to prove the above statement: the argument I know does not work for this. More precisely, I don't know how to prove that $f$ is Lipschitz contiuous also near endpoints $x=0,T$.

I'm glad if you give some hints or proof.

Thank you in advance.

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A function $f$ is called semiconvex if $x\mapsto f(x)+\frac{1}{2}C|x|^2$ is convex for some constant $C>0$, and it is also equivalent to that there exists a constant $C>0$ such that $$ af(x)+(1-a)f(y)\ge f(ax+(1-a)y)+\frac{1}{2}Ca(1-a)|x-y|^2 $$ for all $x,y$ and $a\in[0,1]$.

Answer 1

Just to get an "answer" into the record: The statement is false. A counterexample is given by setting $f=0$ in the interior and $f=1$ at the endpoints.

Answer 2

For a continuous counterexample, consider a function whose graph is a lower-semicircle. It is obviously convex but its slope blows up at each boundary. This tells us we can expect only local Lipschitz continuity.