For $0 \le a \le \frac{1}{2}$ and $0 \le b \le \frac{1}{2}$, does the following hold?
$$ |a - b| \le \Delta \implies |h(a) - h(b) | \le \Delta, $$ where $h(x) = -x \log(x) - (1 - x) \log(1 - x)$ is the binary entropy function and $\Delta$ is a small quantity. I tried taking the lower bound of a binary entropy function like the following: $$4x(1 - x) \le h(x) \le (4x(1 - x))^{(1/\ln(4))},$$ for both $a$ and $b$ but could not go anywhere.
Any pointers would be highly appreciated. Thanks!