pi(x,y) lt;pi(x,a) +pi(a, b) +p,(b,y) lt; From (3) we then have [since f (x) = F(x) and f (y)= F(y)] 3 3 = 81. P2[ F(x), F(y)]lt;3. We conclude...

Solve question no:1of Exercise-6.8,you can use the highlighted theorems from the given attachment.This question is based on uniform continuity which comes under connectedness,completeness and compactness of Metric spaces.

• Attachment 1
• Attachment 2

pi(x,y) <pi(x,a) +pi(a, b) +p,(b,y) <From (3) we then have [since f (x) = F(x) and f (y)= F(y)]33= 81.P2[ F(x), F(y)]<3.We conclude from (4), (5), and (6) thatMogot zredmana Lum Me ToP2 [ F(a), F(b) ]<eprovided only that p, (a, b) <8/3. This shows that F is uniformly continuous on Mi, andthe proof is complete.From 6.8F we see that neither f nor g in 6.8E is uniformly continuous.Exercises 6.81. Given E > 0 find 8 >0 such that[sin x - sina| <e(I.x - a| <8; - 00 <a < co ).[ Hint: Apply the theorem (or law) of the mean to f (x) =sinx.] Deduce that the sinefunction is uniformly continuous on ( - 00, co ).