Does Cesàro continuity imply continuity?

Following the known definition "a function is continuous at $x_0$ iff for any sequence $(u_n)_\Bbb{N}$ converging to $x_0$, the sequence $(f(u_n))_\Bbb{N}$ converges to $f(x_0)$", let's say that a function is Cesàro-continuous at $x_0$ iff for any sequence $(u_n)_\Bbb{N}$ whose Cesàro mean converges to $x_0$, the Cesàro mean of the sequence $(f(u_n))_\Bbb{N}$ converges to $f(x_0)$.

Does Cesàro continuity imply the usual continuity ?

I haven't found any counter example, but I'm also unable to find a way to prove that is is true (if it is) ...

One can prove that if a function $f$ is Cesàro continuous, then $f$ is a linear function and therefore continuous. So Cesàro continuity implies usual continuity.