The exponential function is strictly increasing

Depends on

• The exponential series converges absolutely for every real argument
• The exponential function is smooth and $(\exp)'=\exp$
• The exponential is positive and satisfies $\exp(-x)=1/\exp(x)$
• The mean value theorem, as the case $g(x) = x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a < b$ and differentiable on $(a,b)$ there is $c \in (a,b)$ with $f(b) - f(a) = f'(c)(b-a)$
• The sum of a real power series is continuous at every point strictly inside its interval of convergence

Used by

• Compatible extensions from the finite simple core Corollary
• Lyapunov central limit theorem Corollary
• The exponential is a continuous bijection from ℝ onto (0,∞) Corollary
• A discontinuous positive solution of F(x+y)=F(x)F(y) Counterexample
• Assuming choice, a Hamel-basis additive map transported through exp gives a discontinuous logarithmic function that is not c log Counterexample
• f(x+iy)=eˣ(cos 2y+i sin 2y) is continuous, satisfies f(z+w)=f(z)f(w) and f(1)=e, but is not the standard complex exponential Counterexample
• Feller negligibility cannot be removed from the converse Counterexample
• The exponential is not uniformly continuous on ℝ Counterexample
• Two closed convex sets can have no strong separator Counterexample
• The six hyperbolic functions and their natural domains Definition
• A family containing K₁ is viral for vacuous reasons Example
• A parameter ledger for the high-girth, high-chromatic alteration proof Example
• A positive non-log-convex solution of the Gamma functional equation Example
• Morera proves holomorphy of z↦∫₀¹ tᶻ dt on Rez>1 Example
• xˣ tends to one as x tends to zero from the right Example
• 1+x≤exp(x) for every real x, hence (1-p)ᵐ≤exp(-mp) Lemma
• A large Y-part in a structural comb partition yields the clique-or-stable-set outcome Lemma
• A wide integral geometric layer forces the complete-or-anticomplete property-(*) blockade Lemma
• Absolute real powers are Borel measurable and convex Lemma
• Chernoff bound for independent bernoulli trials Lemma
• Every smaller positive exponent is again an Erdős–Hajnal constant Lemma
• Exponential contraction of projection away from a quasiconvex set Lemma
• Finite simple analytic families and their exact endpoint norms Lemma
• The improper integral of e^-x² over ℝ is finite and positive Lemma
• The plane Gaussian integral equals π by polar coordinates Lemma
• Zero free entire function of exponential type is an exponential Lemma
• The p-functional need not be a norm for 0 < p < 1 Proposition
• Alon–Pach–Solymosi: if H₁ and H₂ have the Erdős–Hajnal property, so does the graph obtained from H₁ by substituting H₂ for a vertex Theorem
• An entire function of polynomial growth is a polynomial Theorem
• Euler's Gamma integral converges exactly for positive real parameters Theorem
• For all positive k,ℓ, some finite graph has girth greater than ℓ and chromatic number greater than k Theorem
• For every t≥1, the class of Kₜ-free graphs has the Erdős–Hajnal property Theorem
• For n≥1 independent random signs, ℙ(|S|≥ t)≤2 exp(-t²/(2n)) for t>0 Theorem
• ker(exp)=2π iℤ, and exp z=exp w exactly when z-w∈2π iℤ Theorem
• ℓᵖ includes into ℓʳ for p < r Theorem
• Morse stability with explicit parameter dependence Theorem
• Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm Theorem
• Regular normalized multiplicative Cauchy equations characterize the exponential Theorem
• Strong law under summable normalized variances Theorem
• The Lᵖ distance for 0 < p < 1 is a complete translation-invariant metric Theorem

…and 5 more results.

Dependency tree · two levels

25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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)