Question

Which of the following given statements is/are correct?

• A. If L.H.D $\ne$ R.H.D, then $f\left(x\right)$ is not differentiable at $x=c$
• B. If a function is differentiable at a point, it is necessarily continuous at the point.
• C. If a function is differentiable at each $x\in R$ then it is said to be every where differentiable.
• D. $\frac{d}{dx}\left(cf\right(x\left)\right)=c\frac{d}{dx}\left(f\right(x\left)\right)$, where $c$ is a constant.

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A If L.H.D $\ne$ R.H.D, then $f\left(x\right)$ is not differentiable at $x=c$
B If a function is differentiable at a point, it is necessarily continuous at the point.
C If a function is differentiable at each $x\in R$ then it is said to be every where differentiable.
D $\frac{d}{dx}\left(cf\right(x\left)\right)=c\frac{d}{dx}\left(f\right(x\left)\right)$, where $c$ is a constant.

Using concept 1 we can say that Option A is correct.

It is mandatory that at a point where a function is differentiable, Left hand limit = Right Hand Limit. Thus, it is continuous at that point. therefore, Option B is correct.

Option C is correct.

Using concept 2, we can say that option D is correct.