Analysis II | Part IB, 2004

Consider a sequence of continuous functions Fn:[1,1]RF_{n}:[-1,1] \rightarrow \mathbb{R}. Suppose that the functions FnF_{n} converge uniformly to some continuous function FF. Show that the integrals 11Fn(x)dx\int_{-1}^{1} F_{n}(x) d x converge to 11F(x)dx\int_{-1}^{1} F(x) d x.

Give an example to show that, even if the functions Fn(x)F_{n}(x) and F(x)F(x) are differentiable, the derivatives Fn(0)F_{n}^{\prime}(0) need not converge to F(0)F^{\prime}(0).

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