The function f(x)=[x] (2 x-1 2 ) , where [.] denotes the greatest integer function, is discontinuous at

The function f(x)=[x] (2 x-1 2 ) , where [.] denotes the greatest…

MCQ

The function $f(x)=[x] \cdot \cos \left(\frac{2 x-1}{2}\right) \pi$, where $[.]$ denotes the greatest integer function, is discontinuous at

A
all irrational numbers $x$.
✓
no $x$.
C
all integer points.
D
every rational $x$ which is not an integer.

✓

Answer

Correct option: B.

no $x$.

(b) : As, $f(x)=[x] \cos \left(\frac{2 x-1}{2}\right) \pi$
For a function to be continuous LHL $=$ RHL $=f(x)$
So, let us check continuity for $x=a$ where $a$ is an integer
L.HL at $x=a$ is given by
$
\begin{aligned}
& \lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{-}}[x] \cos \left(\frac{2 x-1}{2}\right) \pi \\
& =0 \\
&
\end{aligned}
$
RHL at $x=a$ is given by
$
\lim _{x \rightarrow a^{+}} f(x)=\lim _{x \rightarrow a^{+}}[x] \cos \left(\frac{2 x-1}{2}\right) \pi=0
$
Also $f(a)=[a] \cos \left(\frac{2 a-1}{2}\right) \pi=0$
$
\Rightarrow LHL = RHL =f(x), \text { for } x=a
$
So, $f(x)$ will be continuous for all $x$.

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