Give an example to show that the given "definition" of limx→aƒ(x) = L is incorrect.
Definition: For each 0<δ there is an 0<ε such that if 0< l x-a I < δ , then I ƒ(x) - L I < ε .
The Attempt at a Solution
I considered the piece-wise function: ƒ(x) = (0 if x<0) = ( λ if x>0). I then chose an ε such that ε > I λ/2 I , and chose L = λ/2.
It is obviously true that for each positive δ , if 0 < I x-0 I < δ , then I ƒ(x) - λ/2 I < ε . But, by our definition, this means that limx→0ƒ(x) = λ/2 , which is blatantly false.
Is this sufficient? I think that spivak is expecting that I use some sort of "common sense" in my argument such as in my final statement.