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One-sided limit

A limit as x approaches a from the left or from the right in ℝ.
One-sided limit

Let f:E→Rf:E\to\mathbb{R} (or C\mathbb{C}) with E⊆RE\subseteq\mathbb{R}, and let a∈Ra\in\mathbb{R} be a limit point of E∩(a,∞)E\cap(a,\infty). The right-hand limit of ff at aa is LL (written lim⁡x→a+f(x)=L\lim_{x\to a^+} f(x)=L) if

∀ε>0, ∃δ>0 such that ∀x∈E, (0<x−a<δ⇒∣f(x)−L∣<ε).\forall \varepsilon>0,\ \exists \delta>0\ \text{such that}\ \forall x\in E,\ \bigl(0<x-a<\delta \Rightarrow |f(x)-L|<\varepsilon\bigr).

Similarly, the left-hand limit lim⁡x→a−f(x)=L\lim_{x\to a^-} f(x)=L uses the condition 0<a−x<δ0<a-x<\delta (equivalently, −δ<x−a<0-\delta<x-a<0).

One-sided limits are needed for functions defined on half-intervals and for describing jump discontinuities.

Examples:

  • For the step function f(x)=1[0,∞)(x)f(x)=\mathbf{1}_{[0,\infty)}(x), lim⁡x→0−f(x)=0\lim_{x\to 0^-} f(x)=0 and lim⁡x→0+f(x)=1\lim_{x\to 0^+} f(x)=1.
  • For f(x)=1/xf(x)=1/x, lim⁡x→0+f(x)=+∞\lim_{x\to 0^+} f(x)=+\infty and lim⁡x→0−f(x)=−∞\lim_{x\to 0^-} f(x)=-\infty in the extended real sense.
  • If ff is continuous at aa, then both one-sided limits exist and equal f(a)f(a).