By Terence Tao

This can be half considered one of a two-volume creation to genuine research and is meant for honours undergraduates, who've already been uncovered to calculus. The emphasis is on rigour and on foundations. the cloth begins on the very starting - the development of quantity structures and set conception, then is going directly to the fundamentals of research (limits, sequence, continuity, differentiation, Riemann integration), via to strength sequence, numerous variable calculus and Fourier research, and at last to the Lebesgue necessary. those are virtually totally set within the concrete surroundings of the true line and Euclidean areas, even if there's a few fabric on summary metric and topological areas. There are appendices on mathematical common sense and the decimal procedure. the total textual content (omitting a few much less valuable issues) could be taught in quarters of twenty-five to thirty lectures each one. The path fabric is deeply intertwined with the routines, because it is meant that the coed actively research the cloth (and perform considering and writing conscientiously) via proving numerous of the most important ends up in the speculation. the second one variation has been widely revised and up to date.

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**Example text**

2. 5. (Hint: prove the completeness and boundedness separately. For both claims, use proof by contradiction. 3. 7. 4. Let (R, d) be the real line with the standard metric. Give an example of a continuous function f: R--+ R, and an open set V ~ R, such that the image f(V) := {f(x): x E V} of Vis not open. 5. Let (R, d) be the real line with the standard metric. Give an example of a continuous function f : R --+ R, and a closed set F ~ R, such that f(F) is not closed. 6. 9. (Hint: work in the compact metric space (K17 diK1 xKJ, and consider the sets Vn := K1 \Kn, which are open on K1.

We first prove (a). 1 f tfJ g : X ~ R 2 is also continuous at x 0 . 2 the function (x, y) ~---+ x + y is continuous at every point in R 2 , and in particular is continuous at ftiJg(xo). 2. 1. 7 we conclude that f + g : X ~ R is continuous. , g), min(f,g) and cf. 1. 7 to restrict the range of g from R to R\{0}, and then one can argue as before. The claim (b) follows immediately from (a). 0 This corollary allows us to demonstrate the continuity of a large class of functions; we give some examples below.

The set [1, 2] contains both of its boundary points 1, 2 and is hence closed. The set [1, 2) contains one of its boundary poi~ts 1, but does not contain the other boundary point 2, but not the other, so is neither open nor closed. 14. It is possible for a set to be simultaneously open and closed, if it has no boundary. ), and so X is both open and closed. ), and so is both open and closed. In many cases these are the only sets that are simultaneously open and closed, but there are exceptions. For instance, using the discrete metric ddisc, every set is both open and closed!