By Terence Tao

This can be half one among a two-volume advent 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 starts off on the very starting - the development of quantity structures and set idea, then is going directly to the fundamentals of research (limits, sequence, continuity, differentiation, Riemann integration), via to energy sequence, a number of variable calculus and Fourier research, and eventually to the Lebesgue imperative. those are nearly completely set within the concrete environment of the genuine line and Euclidean areas, even though there's a few fabric on summary metric and topological areas. There are appendices on mathematical good judgment and the decimal process. the full textual content (omitting a few much less crucial 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 scholar actively examine the cloth (and perform considering and writing conscientiously) via proving numerous of the main ends up in the idea. the second one version has been largely revised and up-to-date.

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4. 1. 4. 7 (Intermediate value theorem). Let f : X ~ R be a continuous map from one metric space (X, dx) to the real line. Let E be any connected subset of X, and let a, b be any two elements of E. , either f( a) ::; y ::; f(b) or f( a) ;:::: y ;:::: f(b ). Then there exists c E E such that f(c) = y. 5. 1. Let (X, ddisc) be a metric space with the discrete metric. Let E be a subset of X which contains at least two elements. Show that E is disconnected. 2. Let I: X-+ Y be a function from a connected metric space (X, d) to a metric space (Y, ddisc) with the discrete metric.

Z:-+:z:o y-+yo Y-+Yo :z:-+:z:o In particular, we have lim lim f(x, y) = lim lim f(x, y) :z:-+:z:o Y-+Yo y-+yo :z:-+:z:o whenever the limits on both sides exist. (Note that the limits do not necessarily exist in general; consider for instance the function f : R 2 ---+ R such that f(x, y) = y sin~ when xy =f. 7. 10. Let f : R 2 ---+ R be a continuous function. Show that for each x E R, the function y ~---+ f(x, y) is continuous on R, and for each y E R, the function x ~---+ f (x, y) is continuous on R.

By definition of r(y(n)), this implies that r(y(n)) ~ c/2 for all n ~ N. oo r(y(n)) = 0. • Case 2: r 0 > 0. In this case we now have r(y) > ro/2 for all y E Y. ). We now construct a sequence y( 1), y( 2 ), . by the following recursive procedure. We let y(l) be any point in Y. The ball B(y(l), ro/2) is contained in one of the Va and 12. Metric spaces 416 thus cannot cover all of Y, since we would then obtain a finite cover, a contradiction. Thus there exists a point y( 2 ) which does not lie in B(y(l), r 0 /2), so in particular d(y( 2 ), y(l)) ~ ro/2.

### Analysis II (Texts and Readings in Mathematics) by Terence Tao

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