Question

Which of the following is/ are true?

1. If $f\left( x \right)$ be differentiable at the point $x = a$ then the function $f\left( x \right)$ is continuous at that point.
2. If functions $f\left( x \right)$ and $g\left( x \right)$ are continuous at $x = a$ then $f\left( x \right)*g\left( x \right)$ is also continuous at $x = a$.
3. If $f\left( x \right)*g\left( x \right)$ is continuous at $x = a$ then functions $f\left( x \right)$and$g\left( x \right)$ are also continuous at $x = a$.
4. If the function $f\left( x \right)$ is continuous at $x = a$ then the function is also differentiable at $x = a$.

Answer 1

Here, we have to find the condition of the functions. We will use the conditions of the functions such as the continuity and the differentiability of one function or more functions. Using this we will analyze each option and find which statement is true.

Complete step by step solution:
Consider two functions $f\left( x \right)$ and $g\left( x \right)$.
If a function $f\left( x \right)$ is differentiable at a point $x = a$, then the function $f\left( x \right)$ is continuous at the point $x = a$ such that its derivative must be defined.
So, if a function $f\left( x \right)$ is differentiable at a point $x = 1$, then the function $f\left( x \right)$ is continuous at the point $x = 1$.
According to the Algebra of continuous functions, we know that if functions $f\left( x \right)$ and $g\left( x \right)$ are continuous at $x = a$ then $f\left( x \right)*g\left( x \right)$ is also continuous at $x = a$which $*$ is defined under any operation and the converse is also true such that if $f\left( x \right)*g\left( x \right)$ is continuous at $x = a$, then the functions $f\left( x \right)$ and $g\left( x \right)$ are also continuous at $x = a$.
If a function $f\left( x \right)$ is continuous at $x = a$, then it is not necessary that the function should be continuous at the point $x = a$ such that its derivative is not defined.
Therefore, if $f\left( x \right)$ is differentiable at the point $x = a$ then the function $f\left( x \right)$ is continuous at that point.
If functions $f\left( x \right)$and$g\left( x \right)$ are continuous at $x = a$ then $f\left( x \right)*g\left( x \right)$ is also continuous at $x = a$.
If $f\left( x \right)*g\left( x \right)$ is continuous at $x = a$ then functions $f\left( x \right)$ and $g\left( x \right)$ are also continuous at $x = a$.

Thus, option (A), (B) and (C) are the correct answer.

Note:
In order to answer this question, we need to keep in mind the condition of differentiability and continuity. If the right hand derivative is equal to the left hand derivative, then the given function is differentiable at the points. If the right hand derivative is not equal to the left hand derivative, then the given function is not differentiable at the points. If a function is differentiable then the function must be continuous at the same point. So, we can say that differentiability implies Continuity.