Question

Prove/disprove that the set of all continuous functions C[0, 1] defined from the closed unit interval into R,

together with function addition + and function multiplication · is a ring

Answer 1

A ring is an integral Domain if it "has no zero divisors". i.e. if a, $b\isin R$ and ab=0 then a=0 or b=0

To show that your ring is not an integral domain, you need to find two continuous functions f,g

say that are not identically zero, but are such that $f(x)g(x)=0 \ ∀x∈R.$

To prove R is not an integral domain, all you need to do is find an example of zero divisors in R

A simple example is the following: f,g∈R

defined as

$f(x) = \begin{cases} 0 &\text{x } \in (-\infin, 0)\\ x &\text{x } \in[0,\infin) \end{cases}$

and

$g(x) = \begin{cases} -x &\text{x } \in (-\infin, 0)\\ 0 &\text{x } \in[0,\infin) \end{cases}$