Tagged Questions

3
votes
5answers
121 views

Sequence of continuous functions which converges to a continuous limit

Any help with this: construct a sequence of continuous functions defined on $ [0,1] $ which converges pointwise but not uniformly to a continuous limit ? Thank you.
1
vote
3answers
62 views

Math Analysis - Problem with showing sequence of functions is convergent and uniformly convergent

Let $f:\left[\frac{1}{2} ,1\right] \rightarrow \mathbb R$ be a continuous function, $\{g_n\}_{n=1}^{\infty}$ a sequence of functions where $g_n(x) = x^n f(x)$, with $x \in \left[\frac{1}{2} ,1\right]$ ...
1
vote
1answer
58 views

Approximating a function with a polynomials that uniformly converges

Consider f as a continuous function on [a,b]. How can I show that there exists a sequence {$P_n$} of polynomials such that $P_n \rightarrow f$ uniformlyon [a,b] and such that $P_n(a) = f(a)$ for all n ...
1
vote
1answer
128 views

Uniform convergence of continuous functions with Lipschitz limit

Let $K \subset \mathbb R^d$ be a compact. Let $\phi_{\varepsilon} \colon K \rightarrow \mathbb R$ be continuous and converge uniformly to $\phi$. Suppose further that $\phi$ is Lipschitz continuous. ...
3
votes
2answers
370 views

Dini's Theorem. Uniform convergence and Bolzano Weierstrass.

In Spivak's chapter on uniform convergence he asks to prove the following THEOREM Let $\{f_n\}$ be sequence of continuous functions that converge pointwise to $0$ over $[a,b]$. If $0\leq ...