Question

Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous and suppose that $f(x)=0$ whenever $x$ is rational. Show that $f(x)=0$ for every $x$ in $[a, b]$.

   Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous and suppose that $f(x)=0$ whenever $x$ is rational. Show that $f(x)=0$ for every $x$ in $[a, b]$.

Real analysis

N. L. Carothers 1st Edition

Chapter 1, Problem 45

Instant Answer

Step 1

We are given a function \( f:[a, b] \rightarrow \mathbb{R} \) that is continuous on the interval \([a, b]\). Additionally, it is given that \( f(x) = 0 \) for all rational numbers \( x \) within the interval \([a, b]\). Show more…

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Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous and suppose that $f(x)=0$ whenever $x$ is rational. Show that $f(x)=0$ for every $x$ in $[a, b]$.

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