Composite Functions and Uniform Convergence: A Closer Look

[travis0868]

When is the following true?

f(lim_{n\rightarrow\infty}\ g_{n}(x)) =

lim_{n\rightarrow \infty}\ f(g_{n}(x))

Does anyone know of a textbook that discusses this?

 

[Eighty]

Assuming f and g_n are complex-valued, iff f is continuous.

 

[travis0868]

But what if f is the following function:

f(g(x)) = \int^{b}_{a} g(x) dx

If that's the case,

f(lim_{n\rightarrow\infty} g_{n}(x)) = \int^{b}_{a} lim_{n\rightarrow\infty}g_{n}(x) dx

and

lim_{n\rightarrow\infty} f(g_{n}(x)) = lim_{n\rightarrow\infty} \int^{b}_{a} g_{n}(x) dx

These two are equal only when g_{n}(x) is uniformly convergent.