I am a little bit unsure about the following claim. Let $H:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}$ be a mapping of the form $H(x,f(x))$, where $f:[0,1]\rightarrow \mathbb{R}$ is some continuous and nondecreasing function. Suppose there exists a positive constant $K$, independent of $x$ and $f$ such that: $$ \sup_{x\in[0,1]}|H(x,f(x))-H(x,g(x)|\leq K \sup_{x\in [0,1]} |f(x)-g(x)| $$ where $g:[0,1]\rightarrow \mathbb{R}$ is a nondecreasing and continuous function.
The claim is that if the above inequality holds, then $$ |H(x,f(x))-H(x,g(x)|\leq K |f(x)-g(x)| $$ for all $x\in [0,1]$. I want to prove/disprove this claim. I tried a couple of things with no success so far. For instance, I tried to show that $g(x)=\frac{|H(x,f(x))-H(x,g(x))|}{f(x)-g(x)|}$ is bounded above, which would give me the desired result but I am stuck at the point in which I get $$|H(x,f(x))-H(x,g(x))|\leq \sup_{x} g(x) \sup_x|f(x)-g(x)|$$ This looks a lot like the inequality I have, but I do not know how to link $\sup_{x}g(x)$ with $K$.
Any suggestion/comment/help is greatly appreciated it!