Question

Which of the following is are true?

• A. if $f\left(x\right)$ be differentiable at the point $x=a$. Then the function $f\left(x\right)$ is continuous at that point.
• B. If functions $f\left(x\right)$ and $g\left(x\right)$ are continuous at $x=a$, then $f\left(x\right)\ast g\left(x\right)$ is also continuous at $x=a$
• C. If $f\left(x\right)\ast g\left(x\right)$ is continuous at $x=a$ then functions $f\left(x\right)$ and $g\left(x\right)$ are also continuous at $x=a$.
• D. If the function $f\left(x\right)$ is continuous at $x=a$ then the function is also differentiable at $x=a$.

Answer 1

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

The correct options are
A if $f\left(x\right)$ be differentiable at the point $x=a$. Then the function $f\left(x\right)$ is continuous at that point.
B If functions $f\left(x\right)$ and $g\left(x\right)$ are continuous at $x=a$, then $f\left(x\right)\ast g\left(x\right)$ is also continuous at $x=a$
C If $f\left(x\right)\ast g\left(x\right)$ is continuous at $x=a$ then functions $f\left(x\right)$ and $g\left(x\right)$ are also continuous at $x=a$.

A: For a function to be differentiable at some point then function must be continuous at that point and its derivative must be defined (finite)

So if a function is differentiable.

at $x=1\Rightarrow f\left(x\right)$ is continuous at $x=a$

B: If f and g are continuous then according to algebra of continuous functions f,g is also continuous.

C For f,g to be continuous

f and g must be continuous

(refer B)

D: If $f\left(x\right)$ is continuous at $x=a$ It necessarily does not have to be differentiable at $x=a$

$f^{\prime }\left(x\right)$ should also be defined in addition to f being continuous (refer A)

Only A,B,C are given, Answer is A,B,C.