)‌π where [⋅] is greatest integer function and f:R→R ∵ It is a greatest integer function then|we need to check its continuity at x∈I except these it is continuous. Let, x=n where n∈I ‌ Then ‌LHL‌‌=
lim
x→n−
[x−1]‌cos(‌
2x−1
2
)‌π ‌‌=(n−2)‌cos(‌
2n−1
2
)‌π=0 RHL‌‌=
lim
x→n+
[x−1]‌cos(‌
2x−1
2
)‌π ‌‌=(n−1)‌cos(‌
2n−1
2
)‌π=0 and f(n)=0. Here,
lim
x→n−
f(x)=
lim
x→n+
f(x)=f(n) ∴ It is continuous at every integers. Therefore, the given function is continuous for all real x.