Transcription of Solutions to Assignment-3 - UCB Mathematics
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Solutions to Assignment-3 - UCB Mathematics
Solutions to Assignment-31. (a) Letf: (a,b) Rbe continuous such that for somep (a,b),f(p)>0. Show that there exists a >0 such thatf(x)>0 for allx (p ,p+ ).Solution:Let >0 such thatf(p) >0 (for instance one can take =f(p)/2). Sincefiscontinuous, there exists >0 such that|x p|< = |f(x) f(p)|< .In particular, for allx (p ,p+ ),f(x)> f(p) >0.(b) LetE Rbe a subset such that there exists a sequence{xn}inEwith the property thatxn x0/ that there is an unbounded continuous functionf:E :Consider the functionf(x) =1x E, this function is continuous onE. On the other hand, by the hypothesis,limn |f(xn)|= ,and so the function is unbounded (a) Ifa,b R, show thatmax{a,b}=(a+b) +|a b| :Ifa b, then max{a,b}=b.(b) Show that iff1,f2, ,fnare continuous functions on a domainE R, theng(x) = max{f1(x), ,fn(x)}is again a continuous function :Forn= 2, use part(a) to writeg(x) =(f1(x) +f2(x)) +|f1(x) f2(x)| +f2and|f1 f2|are continuous, it follows thatgis also continuous.
and again by the above argument for max of two continuous functions, we see that g k(x) is also continuous. By induction g n(x) = g(x) is also continuous. (c)Let’s explore if …
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