onie mti
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how do i prove that f= mx+c has a local lipschitz property on R
Evgeny.Makarov said:In fact, it has the global Lipschitz property with constant $m$.
Yes. Of course, proving that $f(x)=mx+c$ is Lipschitz by definition is also easy:onie mti said:is it acceptable to say;
suppose that g is differentiable on R.
g'(x)= m
If the derivative is bounded on R, then g is Lip on R and any upper bound for |g'(x)|=m is the lip constant.
Every Lipschitz function is locally Lipschitz.onie mti said:and g' is continuous on R hence g is loc lip.
Evgeny.Makarov said:Yes. Of course, proving that $f(x)=mx+c$ is Lipschitz by definition is also easy:
\[
|f(x_1)-f(x_2)|=|mx_1+c-(mx_2+c)|=|m(x_1-x_2)|=|m||x_1-x_2|.
\]
Every Lipschitz function is locally Lipschitz.