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
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real-analysis
2 Answers
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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 $n \to \infty$.
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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
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Polynomials are dense in $C[0,1]$ by Weierstrass. Those are Lipschitz (in $[0,1]$).
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0... because their derivative is continuous on $[0,1]$, hence bounded. – 2016-05-16