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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.

For the entropy summand $f(x)=-x\log x$, differentiation on $(0,1)$ givesSince $f(x)\to 0$ as $x\downarrow 0$, its continuous extension to $[0,1]$ is a concave function.

Fix $0<v<1$ and define the increment $D_v(t)=f(t+v)-f(t)$ on $[0,1-v]$. In its interior,Thus the increment is decreasing, and its endpoint values are $D_v(0)=f(v)$ and $D_v(1-v)=-f(1-v)$. ThereforeThe cases $v=0,1$ follow directly from the endpoint values.

To obtain the sharper half-interval bound, put $G(v)=f(v)-f(1-v)$. It has $G(0)=G(1/2)=0$, whileHence $G$ is a concave function on $[0,1/2]$ and lies above the chord joining its zero endpoints. Thus $f(v)\ge f(1-v)$ there, proving

Set $N=|\mathcal{X}|$, $d_i=|p_i-q_i|$ and $d={\sum }_i{d}_i=\Vert p-q{\Vert }_1$. If $d=0$, the two probability mass functions agree and the bound is zero. Otherwise every $d_i\le d\le 1/2$. The half-interval entropy summand bound, with $t=\min (p_i,q_i)$ and $v=d_i$, givesBy the triangle inequality, followed by Jensen's inequality for the concave function $f$,This proves the requested finite-alphabet entropy continuity bound: