Continuity of the product of functions

Question

If $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous at $0$ with $f(0)=0$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ is bounded, then the product $fg$ is continuous at $0$. Is this statement true or false? Why?

Answer 1

Hint: if $|g| \le B$, $|fg| \le B |f|$.

Answer 2

$g$ is bounded implies $|g(x)|<M$ for all $x\in\Bbb{R}$. Continuity of $f$ implies, for $\epsilon>0$, there exists $\delta>0$, such that $|f(x)|<\epsilon$ for $|x|<\delta$. Then $|f(x)g(x)|<\epsilon M$ for $|x|<\delta$. So $\lim_{x\rightarrow 0}f(x)g(x)=0$. But $f(0)g(0)=0$. So $fg$ is continuous at $0$.

Answer 3

$f$ is continuous at $x=0$ with $f(0)=0$.

$g$ bounded: $|g(x)| \le M,$ $M$, real, positive.

Let $\epsilon \gt 0$ be given.

There is a $\delta$ such that

$|x| \lt \delta$ implies $|f(x)| \lt \epsilon$.

Let $\varepsilon$ be given.

Choose $\epsilon = \varepsilon/M$.

Then $|x| \lt \delta$ implies

$|f(x)g(x)| \lt M |f(x)| =M\epsilon = \varepsilon$,

hence $fg$ continuous at $x=0$.