GATE | GATE-CS-2007 | Question 1

Consider the following two statements about the function f(x)=|x|

P. f(x) is continuous for all real values of x
Q. f(x) is differentiable for all real values of x 

Which of the following is TRUE?

(A) P is true and Q is false.
(B) P is false and Qis true.
(C) Both P and Q are true
(D) Both P and Q are false.


Answer: (A)

Explanation: A function is continuous if for every value of ‘x’, we have a corresponding f(x). Here, for every x, we have f(x) which is actually the value of x itself, without the negative sign for x < 0.
 

But, the given function is not differentiable for x = 0 because for x < 0, the derivative is negative and for x > 0, the derivative is positive. So, the left hand derivative and right hand derivative do not match.

 

Hence, P is correct and Q is incorrect.
Thus, A is the correct option.

 
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