Hi, i have folowing problem:
>Let [MATH]f:\mathbb{R}^2 \rightarrow \mathbb{R} [/MATH] be a function. Prove that: [MATH]\mathbb{R}^n \ni x \rightarrow(x,f(x)) \in \mathbb{R}^{n+1}[/MATH] is Lipschitz (in Euclid norm) only if [MATH]f[/MATH] is Lipschitz (in Euclid norm).
Is it also true for oethers norms? (norm at [MATH]\mathbb{R}^n[/MATH] may be difference than norm at [MATH]\mathbb{R}^{n+1}[/MATH])
I started from define [MATH]g(x)=(x,f(x))[/MATH] and then [MATH]||g(x)-g(y)||\leq L||x-y||[/MATH] and if i good understand how [MATH]g[/MATH] work now i can write that it is this same as [MATH]||(x,f(x))-(y,f(y))||\leq L||x-y|| \Leftrightarrow ||(x-y,f(x)-f(y))||\leq L||x-y||\Leftrightarrow \sqrt[2]{(x-y)^2+(f(x)-f(y))^2} \leq L\sqrt[2]{(x-y)^2} \Leftrightarrow (x-y)^2+(f(x)-f(y))^2 \leq L(x-y)^2 \Leftrightarrow 1+ \frac{(f(x)-f(y))^2}{(x-y)^2} \leq L [/MATH] and now use that [MATH]f[/MATH] is Lipschitz so we know that [MATH]\frac{(f(x)-f(y))^2}{(x-y)^2} \leq K[/MATH] and now if we take [MATH]L-1=K[/MATH] we finish the proof? Is it correct? But still i have no idea ho to prove part b) "Is it also true for oethers norms? (norm at [MATH]\mathbb{R}^n[/MATH] may be difference than norm at [MATH]\mathbb{R}^{n+1}[/MATH]) "
[MATH][/MATH]
>Let [MATH]f:\mathbb{R}^2 \rightarrow \mathbb{R} [/MATH] be a function. Prove that: [MATH]\mathbb{R}^n \ni x \rightarrow(x,f(x)) \in \mathbb{R}^{n+1}[/MATH] is Lipschitz (in Euclid norm) only if [MATH]f[/MATH] is Lipschitz (in Euclid norm).
Is it also true for oethers norms? (norm at [MATH]\mathbb{R}^n[/MATH] may be difference than norm at [MATH]\mathbb{R}^{n+1}[/MATH])
I started from define [MATH]g(x)=(x,f(x))[/MATH] and then [MATH]||g(x)-g(y)||\leq L||x-y||[/MATH] and if i good understand how [MATH]g[/MATH] work now i can write that it is this same as [MATH]||(x,f(x))-(y,f(y))||\leq L||x-y|| \Leftrightarrow ||(x-y,f(x)-f(y))||\leq L||x-y||\Leftrightarrow \sqrt[2]{(x-y)^2+(f(x)-f(y))^2} \leq L\sqrt[2]{(x-y)^2} \Leftrightarrow (x-y)^2+(f(x)-f(y))^2 \leq L(x-y)^2 \Leftrightarrow 1+ \frac{(f(x)-f(y))^2}{(x-y)^2} \leq L [/MATH] and now use that [MATH]f[/MATH] is Lipschitz so we know that [MATH]\frac{(f(x)-f(y))^2}{(x-y)^2} \leq K[/MATH] and now if we take [MATH]L-1=K[/MATH] we finish the proof? Is it correct? But still i have no idea ho to prove part b) "Is it also true for oethers norms? (norm at [MATH]\mathbb{R}^n[/MATH] may be difference than norm at [MATH]\mathbb{R}^{n+1}[/MATH]) "
[MATH][/MATH]