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
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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 functions?
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.
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 converges uniformly too.
Originally Posted by NonCommAlg not necessarily! in order for that to happen a sufficient condition is that the sequence converges uniformly too. well, this is what I want to show. (df_n) is the differential operator. In fact, is in and is a polynomial function that converges uniformly to . THis is the Weierstrass theorem. Can I say that converges uniformly to ??
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