Could someone please help me with my understanding of the solution by answering my questions written in black ink

Q14. A function f : R - R is said to be periodic on R if there exists a number p > 0 s.t. f(x + p) = f(x) for all a E R. Prove that a continuous periodic function on R is bounded and uniformly continuous on R. what is The Significance of K ? * added? why was Solution. It is easy to deduce f(x) = f(x + kp) for all r E R and for all ke Z. Note that sup f (x) = sup If(x)| IER TE[O,Pl Because [0, p] is a closed and bounded interval, so f is bounded on [0, p] and hence on R. We need to show f is uniformly continuous on R. Let ( > 0. why was This Merval Chosen Because f is uniformly continuous on [-p, 2p], there is o > 0s.t. for all x, y E [-p. 2p]. |x-y)