Finite-alphabet entropy continuity bound ($|H(p)-H(q)|\le d{\log }_2(N/d),d=\Vert p-q{\Vert }_1\le 1/2$)

For probability mass functions on $N$ symbols, set $d_i=|p_i-q_i|$ and $d={\sum }_i{d}_i\le 1/2$. The entropy summand bound gives $|f(p_i)-f(q_i)|\le f(d_i)$. The triangle inequality and Jensen's inequality giveAt $d=0$, use the continuous value zero. This proves uniform continuity of Shannon entropy on a fixed finite alphabet; the alphabet size appears explicitly in the bound.