In $C[0,1]$ prove that the subset of Lipschitz functions is dense. I can't prove it.
In $C[0,1]$ prove that the subset of Lipschitz functions is dense
8
$\begingroup$
real-analysis
2 Answers
7
No need to pull out the heavy guns.. Suppose $f(x) \in C[0,1]$. For a given $n$ let $f_n(x)$ be the piece-wise linear function whose graph connects $(0,f(0))$ to $(1/n, f(1/n))$ to $(2/n, f(2/n))$ to ... to $(1,f(1))$. Then $f_n(x)$ is continuous, it's Lipschitz since it's piecewise linear, and $f_n \to f$ in $C[0,1]$ as $n \to \infty$.
-
0how do you prove that converges to f? – 2012-09-01
-
1Use the uniform continuity of $f(x)$... Given $\epsilon > 0$, if $\delta > 0$ is as in the definition of uniform continuity then if ${1 \over n} < \delta$ then $|f_n(x) - f(x)| < \epsilon$ for all $x$. – 2012-09-02
12
Polynomials are dense in $C[0,1]$ by Weierstrass. Those are Lipschitz (in $[0,1]$).
-
0... because their derivative is continuous on $[0,1]$, hence bounded. – 2016-05-16