Let fn be a sequence of functions in C0 1 R that converges t

Let f_n be a sequence of functions in C([0, 1], R) that converges to f in C([0, 1], R).Examine whether or not fn converges uniformly to f on [0, 1]. Define I: C([0, 1], R) rightarrow R by I (f) = integral_0^1 f(x) dx. Construct a proof that I is continuous on C ([0, 1], R).

Solution

Here f_n converges to f in C([0,1], R).

But we kniw that C([0,1], R) is compact.

By theorem (Any sequenceconverges in compact interval is uniformly convergent),

f_n is uniformly convergent to f in C([0,1], R).

We know that any integrable function in compact interval is differentiable.

Also any differentiable function is continuous.

Therefore I(f) is continuous on the interval.