# Thread: questions on uniform convergence

1. ## questions on uniform convergence

if (x_n) is a sequence that converges uniformly to x and f a continuous fonction

can we say that f(x_n) converges uniformly to f(x)?

thank you

2. Originally Posted by deubelte
if (x_n) is a sequence that converges uniformly to x and f a continuous function.

can we say that f(x_n) converges uniformly to f(x)?

thank you
uniform convergence has meaning for a sequence of functions only! are $\displaystyle x_n$ functions?

3. ahh yes sorry.
to be more specific, this is what I have:
I have a sequence of functions (f_n) that converges uniformly to f.

Now, let d be the derivative operator. It is continuous.
I would like to know if
df_n converge uniformly to df?

thank you.

4. Originally Posted by deubelte
ahh yes sorry.
to be more specific, this is what I have:
I have a sequence of functions (f_n) that converges uniformly to f.

Now, let d be the derivative operator. It is continuous.
I would like to know if
df_n converge uniformly to df?

thank you.
not necessarily! in order for that to happen a sufficient condition is that the sequence $\displaystyle \{f_n' \}$ converges uniformly too.

5. Originally Posted by NonCommAlg
not necessarily! in order for that to happen a sufficient condition is that the sequence $\displaystyle \{f_n' \}$ converges uniformly too.
well, this is what I want to show.

(df_n) is the differential operator.

In fact, $\displaystyle f$ is $\displaystyle C^1$ in $\displaystyle R^n$ and $\displaystyle f_n$ is a polynomial function that converges uniformly to $\displaystyle f$. THis is the Weierstrass theorem.

Can I say that $\displaystyle df_n$ converges uniformly to $\displaystyle df$??