Question

Consider the following statements.
$1$. The function $f\left(x\right)=\left[x\right]$, where $[\cdot ]$ is the greatest integer function defined on R, is continuous at all points except at $x=0$.
$2$. The function $f\left(x\right)=\sin |x|$ is continuous for all $xϵ$R.
Which of the statements is$/$are correct?

• A. $1$ only
• B. $2$ only
• C. Both $1$ and $2$
• D. Neither $1$ nor $2$

Answer 1

Answer: (B)

$2$ only

$Statement1:$

In the graph, we see that

if $x\to 0^{-},$ the value of $f\left(0^{-}\right)=-1$

if $x\to 0^{+}$ the value of $f\left(0^{+}\right)=0$

Since the value of $f\left(0^{-}\right)\ne f\left(0^{+}\right)$ the function is discontinues at x=0

This function is discontinues at every integer.

$Statement2:$

In the graph, we can see the curve is smooth in the throughout the range.Hence it is continues function.