Question: $f:R \rightarrow Rf(x + y) = f(x) + f(y),$ ( [ ] denotes greatest integer function ) equal–...

Question

$f:R \rightarrow Rf(x + y) = f(x) + f(y),$ ( [ ] denotes greatest integer function ) equal–

Answer 1

(i) If x = Rational No. then . πx = Integer π

∴ cos (. πx) = (–1 or 1)

cos²ⁿ ( πx) = + 1

(1 + cos^2m (πx)) = 1 + 1 = 2

(ii) If x Irrational No then $\underset { \mathrm { n } \rightarrow \infty } { \mathrm { h } }$ πx= (Non Integer)π

(∴ – 1 < cos (πx) < 1

⇒ (cos^2m(πx)) = 0 ∴ (less than 1)^∞ = 0

⇒ (1 + cos^2mπx) =1 + 0 = 1

Question: \(f:R \rightarrow Rf(x + y) = f(x) + f(y),\) ( [ ] denotes greatest integer function ) equal–...