Sequence of differentiable functions

real-analysis

A simple counterexample to 1 is the sequence $f_n(x)=\sqrt{(x-1/2)^2+1/n}$, which converges uniformly to non-differentiable function $f(x)=|x-1/2|$.

2 is correct: uniform convergence preserves uniform continuity, and uniform continuity implies Riemann integrability. It follows that 3 and 4 are false.

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