JEE Advance - Mathematics (2017 - Paper 1 Offline - No. 5)
Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function $$f(x) = x\cos (\pi (x + [x]))$$ is discontinuous?
x = $$-$$ 1
x = 1
x = 0
x = 2
Explanation
$$f(x) = x\cos (\pi (x + [x]))$$
At x = 0
$$\mathop {\lim }\limits_{x \to 0} f(x) = \mathop {\lim }\limits_{x \to 0} x\cos (\pi (x + [x]) = 0$$
and f(x) = 0
$$ \therefore $$ It is continuous at x = 0 and clearly discontinuous at other integer points.
At x = 0
$$\mathop {\lim }\limits_{x \to 0} f(x) = \mathop {\lim }\limits_{x \to 0} x\cos (\pi (x + [x]) = 0$$
and f(x) = 0
$$ \therefore $$ It is continuous at x = 0 and clearly discontinuous at other integer points.
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