If f : R → R is a function defined by f \left(\right. x \left.\right) - \left[\right. x \left]\right. cos \textrm{ } \left(\right. \frac{2 x - 1}{2} \left.\right) \pi, where [x] denotes the greatest integer function, then f is :

Engineering

Mathematics

Basic Definition of Continuity

Question

If f : R → R is a function defined by $f (x) - [x] \cos (\frac{2 x - 1}{2}) π$ , where [x] denotes the greatest integer function, then f is :

continuous only at x = 0.

discontinuous only at x = 0.

continuous for every real x.

discontinuous only at non‑zero integral values of x.

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$f : R \underset{}{\to} R$

$f (x) - [x] \cos (\frac{2 x - 1}{2}) π$

[ ] $\underset{}{\to}$ greatest integer function

When $x \in I$ , then f(x) = 0

$[∵ \cos (\frac{2 x - 1}{2}) π = 0 for n \in I]$

For $x \notin I$ then f(x) is product of two continuous function therefore it is continuous.

\ f(x) is continuous for every real x.

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