If a function $f(x,t)$ is globally Lipschitz, does that implies that it is continuously differentiable in $x$?

Question

I know that it is necessary that $df/dx$ continuously exists and it has to be uniformly bounded for $f(x,t)$ to be globally Lipschitz. But, if a function $f(x,t)$ is globally Lipschitz, does that implies that it is continuously differentiable in $x$ ?

Answer 1

The function $f(x,t)=|x|$ is globally Lipschitz,but not continuously differentiable (not even differentiable in $(0,t)$ for any $t$).

What is correct though (I think that's what you mean in the first sentence) that if $f$ is globally Lipschitz and if $\partial f/\partial x$ exists, then this derivative is uniformly bounded. This just follows by writing down the difference quotient and using the Lipschitz assumption.