Legendre transformation is a continuous map

I was reading about an article about the Legendre transformation in convex analysis. It was defined like this :Let $\varphi$ be a continuous function on $[0,+\infty]$ and convex on $\mathbb{R}_{>0}$, we defined the function $\varphi^*$ on $[0,+\infty]$ as :$$ \forall \lambda \in [0,+\infty], \quad \varphi^*(\lambda) := \inf_{x \in [0,+\infty]} (\lambda x + \varphi(x) - \varphi(0))$$$\varphi^*$ is so called the Legendre transform of the function $\varphi$.

So my question is : is $\varphi^*$ continue ?