Question

Let R be the ring of all real valued continuous functions on the closed interval [0, 1]. Let M={f Є R : f(1/3)=0}, show that M is non-empty.​

Answer 1

We want to show that M is non-empty.

• R is the ring of all real valued continuous functions on the closed interval [0,1].
• M = $\{f \in R | f(\frac{1}{3} ) = 0 \}$ .
• Consider the function, $f(x) = 0$ for all $x \in [0,1]$. Since f is a constant real-valued function defined on the interval [0,1], it is continuous and thus belongs in R.
• Also, $f(\frac{1}{3} ) = 0$ . So, f belongs in M.
• Thus M is non-empty