Differential entropy cannot be positive for pdf in (0,1)?

Question

Differential entropy can be non-positive for some functions, but when the domain is an interval of length $1$, I suspect it must be.

Claim. For any probability density function $f$ with domain $(0,1)$,

$$-\int_0^1 f(x) \log f(x)\ dx \leq 0.$$

Is this claim true?

Answer 1

This answer was originally provided as a comment here by user snar.

If $f(x) \geq 0$ on $(0,1)$, then since $\varphi(x) = x \log x$ is convex for $x>0$, by Jensen's inequality $\int_0^1 \varphi(f(x)) dx \geq \varphi \left(\int_0^1 f(x) dx \right) = 0$, so the differential entropy is always non-positive:

$$-\int_0^1 f(x) \log f(x) dx = -\int_0^1 \varphi(f(x)) dx \leq 0.$$