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I don't think you can use the x0 plus minus delta in the bracket (or anywhere), because then the function that's 1 on the rationals and 0 on the irrationals is continuous, because no matter what positive number epsilon is, you can pick delta=7 and x0 plus minus delta is exactly as rational as x0 is so the distance to L is zero, so under epsilon.
You have to say that
whenever |x - 0x|<delta,
|f(x) - L|<epsilon.But I think this is one of my favourite memes.
unless f(x0 ± δ) is some kind of funky shorthand for the set { f(x) : x ∈ ℝ, | x - x0 | < δ }. in that case, the definition would be “correct”.
it’s much more likely that it’s a typo, but analysts have been known to cook up some pretty bizarre notation from time to time, so it’s not totally out of the question.
There's notation for that - (x0 - δ, x0 + δ), so you could say
f(x0 - δ, x0 + δ) ⊂ (L - ε, L + ε)that would be a lot clearer. i’ve just been burned in the past by notation in analysis.
my two most painful memories are:
- in the (baby) rudin textbook, he uses f(x+) to denote the limit of _f _from the right, and f(x-) to denote the limit of f from the left.
- in friedman analysis textbook, he writes the direct sum of vector spaces as M + N instead of using the standard notation M ⊕ N. to make matters worse, he uses M ⊕ N to mean M is orthogonal to N.
there’s the usual “null spaces” instead of “kernel” nonsense. ive also seen lots of analysis books use the → symbol to define functions when they really should have been using the ↦ symbol.
at this point, i wouldn’t put anything past them.
Egregious. I feel your pain.