# question about continuity of a function

everyone can any one solve this problem

Q) show that $$f(x)=\begin{cases} x\sin(1/x)&\text{if }x\ne0\\ 0&\text{if }x=0 \end{cases}$$

is continuous on real number?

by using the definition of continuous

I hope someone can solve. Thanks

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The function $f(x)$ to be continuous at $x_0$ one must require $$\lim_{x\rightarrow x_0-0}f(x) = \lim_{x\rightarrow x_0+0}f(x) = f(x_0)$$ $x\sin \frac 1x$ is even function, so $\lim_{x\rightarrow x_0-0}f(x) = \lim_{x\rightarrow x_0+0}f(x)$, so you just need to find its value. Let's consider $x\rightarrow 0+0$ limit. $\sin(x)$ is bounded by $-1$ and $1$, so $\lim_{x\rightarrow x_0+0}x \sin \frac 1x = 0 = f(0)$