Interactive Real Analysis - part of

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Examples 6.1.3:

  1. Let f(x) = m x + b. Then does the limit of that function exist at an arbitrary point x ?
  2. Let g(x) = [x], where [x] denotes the greatest integer less than or equal to x. Then does the limit of g exist at an integer ? How about at numbers that are not integers ?
  3. In the above definition, does c have to be in the domain D of the function ? Is c in the closure(D) ? Do you know a name for c in terms of topology ?
1. Consider a sequence {xn} converging to c. Then we want to show that f(xn) converges as well. So choose any > 0 and take a positive integer N so large so that

Then we have

| f(xn) - (m c + b) | = | m xn + b - m c - b | = m | xn - c | < m / m =
for n > N. But that means that the sequence f(xn) converges to the number (m c + b).

2. Let g(x) = [x] and assume that c is an integer. Then take the sequence

xn = c + (-1)n / n
Then this sequence converges to c as n approaches infinity, but the sequence g(xn) does not converge at all. Hence, the limit of g(x) does not exist at any integer. On the other hand, if c is not an integer, then the function g(x) converges to the limit L = g(c). Can you prove it ?

3. Recalling our knowledge of topology, we remember that if {xn} is a sequence in a set D converging to c, then c is called an accumulation point of D. Would it be correct to say that a function f(x) with domain D converges to a limit L if for every accumulation point c of D the number L is also an accumulation point of the image of D ?

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