Consider, f (x) = \left{\right. αcosx - \beta \textrm{ }\textrm{ }\textrm{ } ; x \geq 0 \\ \left[\right. \left(sin\right)^{- 1} \left(\right. \frac{2 x}{1 + x^{2}} \left.\right) \left]\right. \textrm{ }\textrm{ }\textrm{ }\textrm{ } ; x < 0. If f (x) is discontinuous at exactly two points x = k1 and x = k2 then [Note : [y] denotes the greatest integer function of y.]

Engineering

Mathematics

Types of Discontinuities

Question

Consider, f (x) = ${\begin{cases} αcosx - β; x \geq 0 \\ [\sin^{- 1} (\frac{2 x}{1 + x^{2}})]; x < 0 \end{cases}$ .
If f (x) is discontinuous at exactly two points x = k₁ and x = k₂ then

[Note : [y] denotes the greatest integer function of y.]

k₁ + k₂ = –2 tan 1

α – β = –1

α – β = 1

k₁ + k₂ = –2 cosec 1

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f (x) is discontinuous at x = k₁ and x = k₂
– (π + 2 tan^–1 x)= –1   and   2 tan^–1 x = –1
   2 tan^–1 x = 1 – π        ⇒ x = – tan= k₂
   tan^–1 x = – $(\frac{π}{2} - \frac{1}{2})$

   x = – cot $\frac{1}{2}$ = k₁
$∴$ k₁ + k₂ = – $(\tan \frac{1}{2} + \cot \frac{1}{2}) = - \frac{1}{\sin \frac{1}{2} \cos \frac{1}{2}} =$ – 2 cosec 1
   f (x) should be continuous at x = 0.
   continuity at x = 0
α – β = –1