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  • Let <img alt="\mathrm{f(x)=\int_0^x t \sin \frac{1}{t} d t}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bf%28x%29%3D%5Cint_0%5Ex%20t%20%5Csin%20%5Cfrac%7B1%7D%7Bt%7D%20d%20t%7D

Let \mathrm{f(x)=\int_0^x t \sin \frac{1}{t} d t}. Then the number of points of discontinuity of the function \mathrm{f(x)} in the open interval \mathrm{(0, \pi)} is

Option: 1

0


Option: 2

1


Option: 3

2


Option: 4

infinite


Answers (1)

best_answer

\mathrm{f^{\prime}(x)=x \sin \frac{1}{x}}. At all points in \mathrm{(0, \pi), f^{\prime}(x)}  has a definite finite value.

\mathrm{\therefore f(x)} is differentiable finitely in \mathrm{(0, \pi)}. As a finitely differentiable function is also continuous, we get \mathrm{f(x)}  is continuous in \mathrm{(0, \pi)}.

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