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The exponential is a continuous bijection from $\mathbb{R}$ onto $(0,\infty)$

Statement

$\exp:\mathbb R\longrightarrow(0,\infty)$ is a bijection.

Facts & Assumptions

Given: The real exponential function.

[L1]

It is strictly increasing (The exponential function is strictly increasing) and continuous (The sum of a real power series is continuous at every point strictly inside its interval of convergence).

[L2]

Its values are positive, and its limits at the two ends are $0$ and $+\infty$ (The exponential tends to $+\infty$ at $+\infty$ and to $0$ at $-\infty$ ).

[L3]

A continuous function on an interval takes every intermediate value (Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a,b]$ takes every value between $f(a)$ and $f(b)$ ).

Proof

technique · direct

1.1

Strict increase gives injectivity.

1.2

Given $y>0$ , [L2] provides $a<b$ with $\exp(a)<y<\exp(b)$ . Applying [L3] on $[a,b]$ gives $c$ with $\exp(c)=y$ .

L1L2L3choose

2.1

Positivity gives the stated codomain, and steps 1.1 and 1.2 give bijectivity.

step 1.1step 1.2L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 105 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)

The exponential is a continuous bijection from R\mathbb{R}R onto (0,∞)(0,\infty)(0,∞)