Question about conjugate of a convex function

Question

I was reading legendre transform and this is what I got Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function then there associates another convex function $f^*$ define by $f^*=sup\{xy-f(x),x\geq 0\}$,
My question is if $f$ is differentiable then we can find $f^*$ but what will happen when $f$ is not differentiable, can someone give an example? Also if someone explains the geometrical interpretation of this or provides some books where I can read about it in more detail then that will be a great help.