If [.] denotes greatest integer function, then at which of the following points $f (x) = [x] sin (\frac{π}{2} ([x] + x))$ is discontinuous?

$x = 0$

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$x = 1$

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$x = 2$

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$x = - 1$

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Solution

The correct options are
A
$x = 0$

C
$x = 2$

D
$x = - 1$

Left-hand limit of $f (x)$ at $x = 1$ is

${lim}_{x \to 1^{-}} [x] sin (\frac{π}{2} ([x] + x)) = 0$

Right-hand limit of $f (x)$ at $x = 1$ is

${lim}_{x \to 1^{+}} [x] sin (\frac{π}{2} ([x] + x)) = 0$

Since, left-hand limit $=$ right-hand limit, the function is continuous at $x = 1$

Given function is discontinuous at all the other integers.

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Similar questions

If [.] denotes greatest integer function and f(x) = [x] ${\frac{s i n \frac{π}{[x + 1]} + s i n π [x + 1]}{1 + [x]}},$ then

f (x) = [x] sin (\frac{π}{[x + 1]}),

[.]

f

f (x) = [x] sin (\frac{π}{[x + 1]})

[.]

f

f (x) = {\begin{matrix} [cos π x], & 0 \leq x \leq 1 | x - 1 | [x - 2], & 1 < x \leq 2 \end{matrix}

([.]

)

f (x) = [x] cos [\frac{2 x - 1}{2}] π

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