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**1. Homework Statement**

Suppose that [itex]A_{n} [/itex] is, for each natural number n, some finite set of numbers in [0,1], and that [itex] A_{n} [/itex] and [itex] A_{m} [/itex] have no members in common if m =/= n. Define f as follows:

[itex]

f(x)=\left\{\begin{array}{cc}1/n,&\mbox{ if }

x \in A{n} \\0, & \mbox{ if }x Not In A_{n} For Any n.\end{array}\right.

[/itex]

Prove that [itex]\lim[/itex] x-a f(x)=0 for all a in [0,1].

Michael Spivak - Calculus Ch5 Q24

## Homework Equations

## The Attempt at a Solution

I've been trying to figure out the existence of this limit, but so far with no luck. I might have misunderstood the problem. Can someone help me out with this?

Update: Just looked a older post on this problem. I think I overlooked some crucial stuff.

So, if we pick a delta value so small that none of the points in A1,A2...An are on the (a-delta, a+delta) interval, except maybe point a itself. f(x) approaches 0 to as x -> a, even if f(a) is defined at some 1/n. Am I being correct?

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