Question

If $f\left(x\right)={\int }_0^{x}t\cos \frac{1}{t}dt$, then the number of points of discontinuity of $f\left(x\right)$ in the inteval $(0,\pi )$ is

Answer 1

We have, $f\left(x\right)={\int }_0^{x}t\cos \frac{1}{t}dt\Rightarrow f^{\prime }\left(x\right)=x\cos \frac{1}{x}$

Clearly, $f^{\prime }\left(x\right)$ exists and is finite in the interval $(0,\pi ).$

Therefore, $f\left(x\right)$ is differentiable in the interval $(0,\pi ).$

Hence, $f\left(x\right)$ is continuous in the interval $(0,\pi ).$