Question:
Which one of the following statements is correct?
$f(x) = \frac{1}{1 + \tan \, x} $
Which one of the following statements is correct?
$f(x) = \frac{1}{1 + \tan \, x} $
Updated On: May 12, 2024
- is a continuous, real-valued function for all $x \in (- \infty,\infty)$
- is discontinuous only at $x = \frac{3\pi}{4}$
- has only finitely many discontinuities on $( - \infty, \infty)$
- has infinitely many discontinuities on $( - \infty, \infty)$
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The Correct Option is D
Solution and Explanation
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) $.
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Notes on Continuity and differentiability
Concepts Used:
Continuity & Differentiability
Definition of Differentiability
f(x) is said to be differentiable at the point x = a, if the derivative f ‘(a) be at every point in its domain. It is given by
Definition of Continuity
Mathematically, a function is said to be continuous at a point x = a, if
It is implicit that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=a exist and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is unspecified or does not exist, then we say that the function is discontinuous.