Question

Which of the following is/are true?
A. If \[f(x)\] be differentiable at the point \[x=a\]. Then the function \[f(x)\] is continuous at that point.
B. If the function \[f\left( x \right)\] and \[g\left( x \right)\] are continuous at \[x=a\]. then \[f\left( x \right)+g\left( x \right)\] is also continuous at \[x=a\].
C. 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 \[x=a\].
D. If the function \[f\left( x \right)\] is continuous at \[x=a\] then the function is also differentiable at \[x=a\]

Answer 1

Hint: From the given question we have to check which of the following are true statements. For the questions of this multiple correct answers we have to check all options. So, for this question we will use the theorem in limits laws and also by taking examples we will solve the question.

Complete step-by-step solution:
Firstly, for the statement A as told in hint we will use the theorem in limits which is as follows.
The theorem is a direct result of limit laws. For instance, to see whether \[f\left( x \right)+g\left( x \right)\] is continuous at \[x=a\], we need to show that \[\displaystyle \lim_{x \to a}\left( f\left( x \right)+g\left( x \right) \right)=f\left( a \right)+g\left( a \right)\].
\[\Rightarrow \displaystyle \lim_{x \to a}f\left( x \right)=f\left( a \right)\text{ }\left( \sin ce\text{ f}\left( x \right)is\text{ continuus} \right)\]
\[\Rightarrow \displaystyle \lim_{x \to a}g\left( x \right)=g\left( a \right)\text{ }\left( \sin ce\text{ g}\left( x \right)is\text{ continuus} \right)\]
\[\begin{align}
  & \Rightarrow \displaystyle \lim_{x \to a}\left( g\left( x \right)+f\left( x \right) \right)=\displaystyle \lim_{x \to a}\text{f}\left( x \right)\text{+}\displaystyle \lim_{x \to a}\text{g}\left( x \right)\text{ }\left( by\text{ limit laws} \right),so \\
 & \Rightarrow \displaystyle \lim_{x \to a}\left( g\left( x \right)+f\left( x \right) \right)=f\left( a \right)+g\left( a \right) \\
\end{align}\]
So, from this we can say option A and option B are true.
Now, for option C we will take an example which is as follows.
Let \[f\left( x \right)=\sin x\] and \[g\left( x \right)=\cot x\]
\[\Rightarrow f\left( x \right).g\left( x \right)=\sin x.\dfrac{\cos x}{\sin x}=\cos x\]
Which is continuous at \[x=0\] but \[\cot x\] is not continuous at \[x=0\].
Therefore option C is false.
Now, for the last option we will take an example and find whether true or false.
If a function is differentiable at a point a, then it is continuous at the point a. but the converse is not true. \[f\left( x \right)=\left| x \right|\] is continuous but not differentiable at \[x=0\].
Therefore, the last option was also false.
Therefore, the solution will be option A and option B are true and option C and D are false.

Note: Students must be having good knowledge in the concept of limits and continuity. Students also know the theorems in limits to solve these types of questions. If students do mistake for example if they write \[f\left( x \right)=\left| x \right|\] is continuous and differentiable at \[x=0\] instead of \[f\left( x \right)=\left| x \right|\] is continuous but not differentiable at \[x=0\] our solution will be wrong.