Consider the following statements.
a) If a function is differentiable at a point $\text{'}p\text{'}$ then it is not continuous at $\text{'}p\text{'}$
b) If a function is not continuous at $x=a,$ then it is not differentiable at $x=a$
c) If $f\left(x\right)=\left|x\right|$ then $f\left(x\right)$ is not differentiable but continuous on $\mathrm{ℝ}$
d) If $f\left(x\right)=x-\left[x\right],$ then $f\text{'}\left(1\right)=1$
Which of the above statements are (is) correct?

We know, if a function is continuous at a point $x=a$ then it is not necessarily differentiable at the point $x=a$ and if a function differentiable at a point $x=a$ then it must be continuous at $x=a.$

$f\left(x\right) =\left|x\right|$ is continuous at every point but it is not differentiable at $x=0$.

$f\left(x\right)=x-\left[x\right]$ is not differentiable at $x=1$ because the greatest integer function is discontinous at the integral points and hence not differentiable at the integral points.

Hence, only (b) and (c) are true.