Suppose you have this two Lipschitz continuous functions:

$f_1$ ,with constant $L_1$ and $f_2$ with constant $L_2$.

I have to prove that $f_2 - f_1$ is Lipschitz continuous with constant $L_1+L_2$.

I did like this:



By the triangle inequality:


$\leq L_2|y-x|+|-y+x|$

But I´m having problems in this last line,above...I should have $|y-x|$ in both,isn´t it?What´s wrong?how to proceed?Thanks?

  • 1
    $\begingroup$ You are missing at $L_1$ on the last line of your formula. Also, $|z| = |-z|$, so $|-y+x| = |x-y| = |y-x|$. $\endgroup$ – copper.hat Jul 12 '12 at 19:42

One of the properties of a norm tells you that $|\lambda u|=|\lambda||u|$ for every scalar $\lambda$ and every vector $u$. Therefore $|-f_1(y)+f_1(x)|=|f_1(y)-f_1(x)|\le L_1|y-x|$ .


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