If f: R arrow R is a function defined by f(x)=[x] ≤ft( \; 2x-1/2 ) π, where [x] denotes the greatest integer function, then f is [AIEEE 2012]

Correct Answer is : continuous for every real xxx

discontinuous only at non-zero integral values of x

Correct Answer is : continuous for every real xxx

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AIEEE 2012 Solved Paper

Section: Mathematics

Question : 85 of 89

Marks: +1, -0

f: R \rightarrow R

f(x)=[x] \cos \left( \; \frac{2x-1}{2} \right) \pi,

[x]

f

Solution:

Let

f(x)=[x] \cos \left( \; \frac{2x-1}{2} \right)

Doubtful points are

x=n, n \in I

\text{L.H.L.} = \lim\limits_{x \to n^{-}} [x] \cos \left( \; \frac{2x-1}{2} \right) \pi

= (n-1) \cos \left( \; \frac{2n-1}{2} \right) \pi = 0

(As

[x]

is the greatest integer function )

R.H. L.

= \lim\limits_{x \to n^{+}} [x] \cos \left( \; \frac{2x-1}{2} \right) \pi = n \cos \left( \; \frac{2n-1}{2} \right) \pi = 0

Now, value of the function at

x=n

f(n)=0

Since,

\text{L.H.L.} = \text{R.H.L.} = f(n)

\therefore \ f(x)=[x] \cos \left( \; \frac{2x-1}{2} \right)

is continuous for every real

x

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