Find all function $f$ such that, for any sequence $(x_n)$ Cesàro-convergent, the sequence $(f (x_n))$ is also Cesàro-convergent Let's say that a function $f$ 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 $f(x_0)$. At first, I was interested in the following question

Find all functions $f : \mathbb{R} \to \mathbb{R}$ Cesàro-continuous.

I realized that it is a classic question and the answer is that $f$ must be affine. Now I'm interested in this question

Find all function $f : \mathbb{R} \to \mathbb{R}$ such that, for any sequence $(x_n)$ Cesàro-convergent, the sequence, the sequence $(f(x_n))$ is also Cesàro-convergent?

It seems to me that the two questions are equivalent, but I do not know if it's true and how to prove it.
I need help. The idea of the question came to me by this question.
 A: Let $ {\left({u}_{n}\right)}_{n \in  \mathbb{N}}$ Cesaro-converge to $\ell $, and let $ {\left({v}_{n}\right)}_{n \in  \mathbb{N}}$ be the constant sequence $ {\left(\ell \right)}_{n \in  \mathbb{N}}$. The sequence $ f \left({v}_{n}\right)$ Cesaro-converges to $ f \left(\ell \right)$. Let us suppose that $ f \left({u}_{n}\right)$ Cesaro-converges to a different limit $ {\lambda}$ and let $0  <  {\varepsilon}  <  \left|f \left(\ell \right)-{\lambda}\right|/3$.
Any sequence $ {w}_{n}$ built by interleaving consecutive terms taken from the sequences $ {u}_{n}$ and $ {v}_{n}$ also Cesaro-converges to $\ell $. We build such a sequence by alternating the two following steps
$\ $
A) Pick items from the sequence $ {u}_{n}$ until the average of the terms $ f \left({w}_{n}\right)$ chosen so far is in $\left[{\lambda}-{\varepsilon} , {\lambda}+{\varepsilon}\right]$
B) Pick items from the sequence $ {v}_{n}$ until the average of the terms $ f \left({w}_{n}\right)$ chosen so far is in  $\left[f \left(\ell \right)-{\varepsilon} , f \left(\ell \right)+{\varepsilon}\right]$
The sequence $ {w}_{n}$ Cesaro converges but $f \left({w}_{n}\right)$ does not, a contradiction.
It follows that $f$ must be Cesaro-continuous.
