Question

If f:R-->R is a function defined by f(x)=[x]cos(2x-1/2)pi ,where [x] denotes the greatest interger function then f is
1) continuous for every real x
2) continuous only at X is equals to zero
3) discontinuous only at non-zero integer values of x
4) continuous only at x is equals to zero ​

Answer 1

Answer:

f(x)=[x]cos

2

(2x−1)

π

=[x]cos(x−

2

1

)π

=[x]cos(πx−

2

π

)=[x]sinπx

For all Non Integral values of x,f(x) is continous.

Let n be an integer.

lim

x→ n

+

f(x)=0

lim

x→ n

−

f(x)=0

∴f(n)=0

∴f(x) is continuous for every real x.