Can gradient of a function be Lipschitz continuous but the function is not Lipschitz?

Question

Can gradient of a function be Lipschitz continuous but the function is not Lipschitz?

Any figurative example where gradient is Lipschitz but the function is not Lipschitz possible would be really intuitive, can someone illuminate me? Thank you so much in advance for your help.

Answer 1

Consider the function $$f(t):=t^2\qquad(-\infty<t<\infty)\ .$$ This function is not Lipschitz continuous on its domain, but $$\nabla f(t)=f'(t)=2t$$ is.