Exercise 4.6.4

Question

Given $f: A ightarrow \mathbf{R}$ and a limit point $c$ of $A, \lim _{x ightarrow c} f(x)=L$ if and only if
  $$
  \lim _{x ightarrow c^{-}} f(x)=L \quad 	ext { and } \quad \lim _{x ightarrow c^{+}} f(x)=L .
  $$
  Supply a proof for this proposition.

Ulisse Mini · Accepted Answer

Let $\epsilon > 0$, pick $\delta_1$ so $0 < x-c < \delta_1$ implies $|f(x)-L|<\epsilon$, pick $\delta_2$ so $0 < c-x < \delta_2$ implies $|f(x)-L|<\epsilon$. Finally, set $\delta = \min\{\delta_1, \delta_2\}$ to get $|f(x)-L|<\epsilon$ when $0 < |x-c| < \delta$, as desired.

Exercise 4.6.4

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Exercise 4.6.4

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