Which of the following statements are correct?

(A) If $f: R \rightarrow R$ then $f(x)=|x|$ is continuous everywhere.
(B) If $f: R \rightarrow R$ then $f(x)=|x|$ is continuous everywhere but not differentiable at $x=0$.
(C) Let $f: R-\{0\} \rightarrow R$ then $f(x)=\frac{1}{x}$ is continuous everywhere.
(D) Let $f: R \rightarrow R$ then $f(x)=|x-1|+|x-2|$ is continuous everywhere but not differentiable at exactly 2 points.
(E) If $f: R \rightarrow R$ then $f(x)=\cot x$ is continuous everywhere.

Choose the correct answer from the options given below :

(A), (B), (C), (D) only

A. → Correct as there is no discontinuity

B. → Correct

C. → Correct

D. → Correct

E → Incorrect as $\cot x = \frac{\cos x}{\sin x}$  so for sin x = 0 is discontinuous i.e. at $x = n\pi$ it is discontinuous