I am trying to find a mistake in the logic below. Something definitely doesn't match up.
Let's consider function
$f(x)=\|x-c\|_2^2$.
It is Lipshitz Continuous (L-continuous) with $L=1$. Gradient of this function is equal to $\nabla f(x) = 2 (x-c)$. Therefore, using the property of L-continuous function, it should be correct that
$\|\nabla f(x)\|_2\leq L = 1$.
However, it is definitely not true, for example for $x=0$ and $c=2$. Could anyone explain it to me, please?
Proof that two norm function is L-continuous can be found here https://mathhelpboards.com/analysis-50/euclidean-norm-lipschitz-continuous-d-amp-k-example-1-3-5-a-23484.html