Give examples of the following no proofs required A function

Give examples of the following, (no proofs required): A function differentiable everywhere but with discontinuous derivative. Two functions F(x) and f(x) such that F(x) = integral_0^x f(s) ds but F\' notequalto f. Find two sequences of functions {f_n} n N, {g_n} n N converging uniformly on a set such that the product {f_n g_n} n N does not converge uniformly on that set. A sequence of functions discontinuous everywhere which converges uniformly to a continuous function. A sequence of functions {f_n} n N which converges pointwise to f but where the convergence is not uniform.

Solution

f(x)={x2sin(1/x);if x0

0 ;if x=0.

f(x)={x2(1x)2sin(1x(1x));if 0<x<1else.

0 ;else.