# To prove a set is a subring but not an ideal.

Problem

Let $R$ be the ring of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Let $A=\{f \in R \mid f(0)\text{ is an even integer}\}$. Show that $A$ is a subring of $R$, but not an ideal of $R$.

Silly doubt

Under what binary operations this is a ring? Usually in the case of group it is composition. How to proceed here?

If that part is clear, then this problem can be solved.

The operations are addition and multiplication of functions, i.e. $$(f+g)(x)=f(x)+g(x) \\ (fg)(x)=f(x)g(x).$$