Limits at Infinity and Monotone Functions

Two features of the real line have no metric-space analogue: the line has two ends, and it is ordered. The ends give the limit of $f (x)$ as $x \to + \infty$ or $- \infty$ , which makes asymptotes precise. The order gives the class of monotone functions, whose structure alone forces their one-sided limits to exist and restricts their discontinuities to countably many jumps. The two threads meet in the continuity of inverse functions.

Limits at infinity

Infinity is not a real number, but it can play the role of a cluster point: a set clusters at $\infty$ when it contains arbitrarily large elements.

The threshold $M$ plays the role $δ$ played at a finite point: it names how far out you must go before $f$ stays within $ϵ$ of $L$ . When such an $L$ exists the line $y = L$ is a horizontal asymptote. As usual the limit, if it exists, is unique.

A horizontal asymptote: past the threshold M the graph stays inside the epsilon-band around L, and it never leaves again as x grows.

A sequential characterization holds here too: $lim_{x \to \infty} f (x) = L$ iff $f (x_{n}) \to L$ for every sequence $x_{n} \to \infty$ in $S$ . Through it, the algebra of limits carries over with the cluster point allowed to be $\pm \infty$ .

Infinite limits

It is convenient to name divergence that is orderly rather than oscillatory.

Limits compose under a continuity hypothesis exactly as at finite points; for instance $lim_{x \to \infty} e^{- x^{2} + x} = 0$ , since $- x^{2} + x \to - \infty$ and $e^{y} \to 0$ as $y \to - \infty$ .¹

Monotone functions

A function that only rises, or only falls, cannot oscillate, and this restriction alone guarantees one-sided limits and restricts the discontinuities to jumps.

Since $- f$ reverses the direction, results for increasing functions convert directly to decreasing ones. The key fact is that monotonicity forces one-sided limits to exist, computed as suprema and infima of the values on one side.

All one-sided limits of a monotone function exist whenever they make sense. Consequently, for $c$ interior to the domain,

x \to c^{-} lim f (x) \leq f (c) \leq x \to c^{+} lim f (x),

and $f$ is continuous at $c$ exactly when these two coincide. A discontinuity of a monotone function is therefore always a jump: a gap between the left supremum and the right infimum.

At a jump of an increasing function the left limit is the supremum of earlier values and the right limit the infimum of later ones; the gap between them is the height of the jump, with f(c) sitting inside it.

The floor function: constant on each half-open interval between integers, jumping by one at each integer, where the value joins the upper step.

Countably many discontinuities

The jumps of a monotone function cannot pile up too densely, because each jump encloses an open gap and the gaps are disjoint.

A monotone function can still be discontinuous on a dense set such as the rationals, but never on more than a countable set.

An increasing function's jumps enclose disjoint open gaps on the y-axis; choosing one rational inside each gap injects the jumps into the rationals, so there are at most countably many.

Continuity, images, and inverses

For monotone functions on an interval, continuity is equivalent to a purely set-theoretic condition on the image, via the intermediate value theorem.

The order structure also yields continuity of inverses. A strictly monotone function is injective, so it has an inverse on its range.

When $f (I)$ happens to be an interval, the image characterization applies to both $f$ and $f^{- 1}$ : an onto strictly monotone map between two intervals is a continuous bijection with continuous inverse, a homeomorphism of intervals.

This continuity of the inverse is the topological half of the inverse function theorem: once differentiability is added, the inverse of a differentiable strictly monotone function is differentiable wherever $f^{'} \neq = 0$ , with derivative the reciprocal of $f^{'}$ .

Lebl, Basic Analysis I, §3.5 — Definitions 3.5.1 and 3.5.6, Lemma 3.5.5, Proposition 3.5.8, and Examples 3.5.3–3.5.9 (limits at infinity, infinite limits, compositions). ↩ ↩² ↩³
Lebl, Basic Analysis I, §3.6 — Definition 3.6.1, Proposition 3.6.2, Corollaries 3.6.3–3.6.4, and Proposition 3.6.6 (monotone functions, one-sided limits, countable discontinuities, continuity of inverses). ↩ ↩² ↩³ ↩⁴ ↩⁵