Understanding notation in ring homomorphism I'm doing a problem checking whether it's ring homomorphism or not.

$\phi_0 : C^1(\mathbb{R}) \to C(\mathbb{R}), f(x) \to f(0)$

What is $C^1$? and how does $f$ work? Don't I need the definition of $f$?
 A: $C^n(\Bbb R)$ is the set of real-valued functions with domain $\Bbb R$ which are differentiable $n$ times and such that $f^{(n)}$ is continuous.
$\phi_0:C^1(\Bbb R)\to \Bbb R$, $f\to f(0)$ is a notation for $$\phi_0:C^1(\Bbb R)\to\Bbb R\\ \phi_0(f)=f(0)$$
I think modern authors, when they use an arrow instead of $g:A\to B,\ g(x)=\text{stuff}$, prefer to write $g:A\to B,\ x\mapsto \text{stuff}$.
A: From https://en.wikipedia.org/wiki/Continuous_function, $C^1(\Bbb{R})$ is the set of all continuous functions on $\Bbb{R}$ that have continuous first derivatives.
For any continuous function $f$ on $\Bbb{R}$, $\phi_0$ will send $f$ to the image of $0$ under $f$.
I think examples of $f$ might help you.
(1) If $f(x)=2x+3$, then $\phi_0(f)=f(0)=3$.
(2) If $f(x)=\sin x$, then $\phi_0(f)=f(0)=0$.
Note that $f$ is not fixed. It is just an arbitrary element in the set $C^1(\Bbb{R})$. Hence you do not need the definition of $f$.
However, you need to know how the addition and multiplication are defined on $C^1(\Bbb{R})$. For $f,g\in C^1(\Bbb{R})$,
$$(f+g)(x)=f(x)+g(x), \text{ and } (f\cdot g)(x)=f(x)g(x)$$
where $x\in \Bbb{R}$. By using these two definitions, it can be easily shown that $\phi_0$ is a ring homomorphism.
