Question
Download Solution PDFIf f : R → R is a function defined by f(x) = [x – 1] cos ((2x – 1)/2)π, where [.] denotes the greatest integer function, then f is:
- discontinuous only at x = 1
- discontinuous at all integral values of x except at x = 1
- continuous only at x = 1
- continuous for every real x
Answer (Detailed Solution Below)
Option 4 : continuous for every real x
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Detailed Solution
Download Solution PDFConcept:
A function f(x) is continuous at x = a if \(\lim_{x\to a^-}\ f(x)=\lim_{x\to a^+}\ f(x)=\lim_{x\to a}\ f(x)\)
Calculation
Doubtful points are x = n, n∈I
LHL = \(\lim _{x \rightarrow n^{-}}[x-1] \cos \left(\frac{2 x-1}{2}\right) \pi=(n-2) \cos \left(\frac{2 n-1}{2}\right) \pi=0\)
RHL = \(\lim _{x \rightarrow n^{+}}[x-1] \cos \left(\frac{2 x-1}{2}\right) \pi=(n-1) \cos \left(\frac{2 n-1}{2}\right) \pi=0\)
⇒ f(x) = 0 ∀ x ∈ R
∴ f is continuous for every real x.
The correct answer is Option 4.
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