Showing a certain type of function on $\mathbb R ^d$ is Lipschitz?

I want to prove the following:

Assume that $f:\mathbb R ^d \to \mathbb R$ is continuous, convex and $|f(x)|\leq a+b|x|$. Then $f$ is Lipschitz.

I thought it would follow immediately from the assumptions, but I could not get the Lipschitzness no matter what I tried. Any help would be greatly appreciated.