Convex conjugate of x log x ($f^{\ast }(p)=e^{p-1}$)

For $f(x)=x\log x$ on $x>0$, strict convexity and the stationary condition $p=1+\log x$ give the displayed convex conjugate for every real $p$. Biconjugation recovers $f$ on its original positive domain. Its closed extended-real extension has value zero at zero and infinity on negative arguments.