# Sine function is Uniformly Continuous Assignment | Assignment Help Services

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.

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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)]33= 81.P2[ F(x), F(y)]&lt;3.We conclude from (4), (5), and (6) thatMogot zredmana Lum Me ToP2 [ F(a), F(b) ]&lt;eprovided only that p, (a, b) &lt;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 &gt; 0 find 8 &gt;0 such that[sin x – sina| &lt;e(I.x – a| &lt;8; – 00 &lt;a &lt; co ).[ Hint: Apply the theorem (or law) of the mean to f (x) =sinx.] Deduce that the sinefunction is uniformly continuous on ( – 00, co ).

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