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Let $(X,\rho)$ be a metric space. Suppose $f$ to be a real-valued funtion on $X$. A function $w: \left[0,\infty\right)\rightarrow \left[0,\infty\right)$ is said to be a modulus of continuity of the function $f$, if $|f(x)-f(y)|\leqslant w(\rho(x,y))\;\;\; (x,y\in X).$

My question is if the following sentence is true:

Assume that $K$ be a compact subest of X and let $f:K\rightarrow \mathbb{R}$ be a continuous function. Then $f$ has a modulus of continuity $w$ which is increasing, concave and $\lim_{t\to 0} w(t)=w(0)=0$.

I know that the function $w$ given by $w(t)=\sup_{\rho(x,y)=t} |f(x)-f(y)|\;\;\; (t\geq 0),$ is a modulus of continuity of such $f$ and $\lim_{t\to 0} w(t)=w(0)=0$, and then we may set $w_1(t)=\sup_{s\leq t} w(s)$ which is also increasing, but is possible to obtain the concavity too?

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    You can take the concave envelope of $\omega$, I wrote a brief explanation [here](http://calculus7.org/2012/03/05/uniform-continuity-done-right/)2012-09-01

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