Example of a function which is nearly a linear transformation

Let $V$ and $W$ be vector spaces over the field $\mathbb{F}.$ Let $f$ be a function from $V$ to $W.$ Now $f$ will be called a linear transformation if

\begin{align}
&\tag1 f(\alpha + \beta) = f(\alpha) + f(\beta)\,\, \forall \alpha,\beta \in V \\
&f(c\alpha) = cf(\alpha)\,\, \forall c\in \mathbb{F}\tag2

\end{align}

I am interested in finding examples of functions where :

(a) the first condition $(1)$ fails and second condition $(2)$ holds

(b) the second condition $(2)$ fails and first condition $(1)$ holds

I have two examples for the (b) part:

Consider $f : M_{n\times n}(\mathbb{C}) \rightarrow M_{n\times n}(\mathbb{C}) $ with the mapping $A \to A^*$ where $A^*$ is the conjugate transpose of $A.$

Consider $f : \mathbb{C} \rightarrow \mathbb{C}$ wih the mapping $z \to \overline{z}.$

So far I haven't been able to find an example for (a).Please help me wind this up. Also if you find more examples for (b), please list them too.

Answer

Let the field be the the field of complex numbers and the vector space be the vector space of complex numbers over the field of complex numbers.
The function $f(z)=Re(z)$ satisfies $$f(z+w)=f(z)+f(w)$$ but fails to satisfy, f(Cz)=Cf(z) for a complex number C.

Let consider the vector space of complex numbers over the real field.

Define $f(z)=(\text {Sgn} (Re(z))|z|$,where Sgn stands for the signum function.
Then $$f(z+w)=f(z)+f(w)$$ fails but $f(cz)=cf(z)$ holds.