While reading Csiszár & Körner's "Information Theory: Coding Theorems for Discrete Memoryless Systems", I came across the following argument:
Since $f(t) \triangleq -t\log t$ is concave and $f(0) = 0$ and $f(1) = 0$, we have for every $0 \leq t \leq 1-\tau$, $0 \leq \tau \leq 1/2$, \begin{equation}|f(t) - f(t+\tau)| \leq \max (f(\tau), f(1-\tau)) = -\tau \log\tau\end{equation}
I can't make any progress in seeing how this bound follows from the properties of $f(t)$. Any insights would be greatly appreciated.