Prove that $f(x) = 0$ for all $x ∈ R$.

Question

Let $f(x)$ be a continuous function such that $f(r) = 0$ for all rational numbers r. Prove that $f(x) = 0$ for all $x ∈ R$.

Use the property that the rationals are dense in the reals. – Raskolnikov Dec 04 '14 at 22:35 — Raskolnikov, Dec 04 '14 at 22:35
Note that rational numbers are totally disconnected. – Tacet Dec 04 '14 at 22:36 — Tacet, Dec 04 '14 at 22:36

score 4 · Answer 1 · answered Dec 04 '14 at 22:34

4

By definition of the real numbers, any real number is limit of a sequence of rational numbers. Continuous functions preserve limits.

answered Dec 04 '14 at 22:34

1k5

This answer is commensurate with the level of the question and very probably the kind of solution the lecturer is looking for. – Simon S Dec 05 '14 at 00:15

score 0 · Answer 2 · edited Apr 13 '17 at 12:21

0

I feel like the following equivalent definition of continuity is very suitable here:

$$f \textrm{ is continuous}\ \iff f(\overline{M})\subset \overline{f(M)} \textrm{ for all}\ M\subset X $$

Now choose $M=\mathbb Q$ and you are done.

A proof of the above alternative definition of continuity can be found here.

edited Apr 13 '17 at 12:21

Community

answered Dec 04 '14 at 23:06

Marm

2 Answers2