Recall the theorem in class that said that a function f:(M,dM) rightarrow (N, dN
ID: 3077697 • Letter: R
Question
Recall the theorem in class that said that a function f:(M,dM) rightarrow (N, dN) is continuous if and only if for all sequences (pn) in M for which pn rightarrow p. f(pn) rightarrow f(p). Consider the following statement: A function f : (M,dm) rightarrow (N, dN) is continuous if and only if for all convergent sequences (pn) in M, the sequence f(pn) is convergent in N. How is this statement different from the statement in the theorem we had in class? Prove that this statement is true.Explanation / Answer
As we’ve seen, when our target is a higher-dimensional real space continuity is the same as continuity in each component. But what about when the source is such a space? It turns out that it’s not quite so simple. One thing, at least, is unchanged. We can still say that f:mathbb{R}^m ightarrowmathbb{R}^n is continuous at a point ainmathbb{R}^m if limlimits_{x o a}f(x)=f(a). That is, if we have a sequence left{a_i ight}_{i=0}^n of points in mathbb{R}^m (we only need to consider sequences because metric spaces are sequential) that converges to a, then the image of this sequence left{f(a_i) ight}_{i=0}^n converges to f(a). The problem is that limits themselves in higher-dimensional real spaces become a little hairy. In mathbb{R} there’s really only two directions along which a sequence can converge to a given point. If we have a sequence converging from the right and another sequence converging from the left, that basically is enough to establish what the limit of the function is (and if it has one). In higher-dimensional spaces — even just in mathbb{R}^2 — we have so many possible approaches to any given point that in order to avoid an infinite amount of work we have to use something like the formal definition of limits in terms of metric balls. That is The function f:mathbb{R}^n ightarrowmathbb{R} has limit L at the point a if for every epsilon>0 there is a delta>0 so that delta>lVert x-a Vert>0 implies lvert f(x)-L vertRelated Questions
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