Examples 6.1.3:
- Let f(x) = m x + b. Then does the limit of that function exist at an arbitrary point x ?
- 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 ?
- 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 ?
- | xn - c| < / m for n > N
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 / nThen 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 ?