Proof Verification: $x\sin(1/x)$ is continuous on $(0,1)$

Question

Prove that $x\sin(1/x)$ is continuous on $(0,1)$

Proof:

Let $\epsilon>0$ be given. Then $\exists \delta>0$ such that if $x,c \in (0,1)$ and if $|x-c|<\delta \implies |f(x)-f(c)|<\epsilon$

Then, $|x\sin(\frac{1}{x}) - c\sin(\frac{1}{c})|\le |x|\frac{1}{x}| -c|\frac{1}{c}||=|1-1|=0<\epsilon$

Then, $x\sin(1/x)$ is continuous on $(0,1)$

Note: Inequality used is $\sin(x) \le |x|$

Can anyone please verify this proof for me and let me know if there are any loopholes in my argument?

Thank you.

Answer 1

Does the inequality $$| x\sin \frac{1}{x} - c\sin \frac{1}{c} | \leq | |x||\frac{1}{x}| - |c||\frac{1}{c}| | $$ hold ?

Note: $ | |x|-|y| | \leq |x-y|$