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.