Question

Which one of the following statements is correct?
$f(x)=\frac{1}{1+\tan x}$

• A. is a continuous, real-valued function for all $x\in (-\infty ,\infty )$
• B. is discontinuous only at $x=\frac{3\pi }{4}$
• C. has only finitely many discontinuities on $(-\infty ,\infty )$
• D. has infinitely many discontinuities on $(-\infty ,\infty )$

Answer 1

Answer: (D)

tan x is not continuous at $x=\frac{\pi }{2},3\frac{\pi }{2},5\frac{\pi }{2}$etc...
So, tan x has infinitely many discontinuities on $(-\infty ,\infty )$
$\Rightarrow f(x)=\frac{1}{1+\tan x}$ has infinitely many discontinuities on $(-\infty ,\infty )$.