- #1

- 1,462

- 44

## Homework Statement

##f(x) = x \sin (\frac{1}{x})## for ##x \ne 0## and ##f(0) = 0##. Prove that this function is continuous at 0.

## Homework Equations

## The Attempt at a Solution

First, I need to look at the quantity ##|f(x) - f(0)|##. However, I am not completely sure how to proceed. I would think that we substitute ##x \sin (\frac{1}{x})## for ##f(x)##, but is this justified? Why can we guaruntee in this case that ##x## won't assume the value of 0 and hence be 0 rather than ##x \sin (\frac{1}{x})##?