= Weierstrass approximation does not preserve derivatives
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On $[0,1]$, the <polynomials> $p_n(x)=x^n/n$ converge in <supremum norm> to the zero <function>, since $\|p_n\|_\infty=1/n$. But $p_n'(1)=1$ for every $n$, whereas the zero <function> has left <derivative> zero at that endpoint. Thus <uniform convergence> of function values does not imply <uniform convergence> of <derivatives>. This does not contradict <uniform convergence of Bernstein polynomial derivatives>, which has a particular approximation construction and additional <differentiability> hypotheses.
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