Is there a function over $\mathbb{Z}_p$ that is never linear?

Question

Let $p$ be a prime. I wonder if there is a function $f$ that satisfies the following rule:

Whenever $$z_1 + \dots + z_c \equiv cx \mod p$$ (where $1 < c < p$, and $z_j, x \in \mathbb{Z}_p$)

Then: $$f(z_1) + \dots + f(z_c) \not \equiv cf(x) \mod p$$

Does such a function exist?

Answer 1

As you have written it, it is not possible.

Consider any function $f$ on $\mathbb Z_p$, and let $f(0) = n$ ($0 \le n < p$). Now $\underbrace{0 + 0 + \cdots + 0}_c = c \times 0$, and $\underbrace{f(0) + f(0) + \cdots + f(0)}_c = cn = cf(0)$.

Your definition does not seem like the usual definition of linearity anyway. If it's really linearity you want, I think you should go with Zev Chonoles's definition.