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

• Finite-class empirical-risk selection with exponential weights Algorithm
• A sample-size consequence for compression schemes Corollary
• Compatible extensions from the finite simple core Corollary
• Lyapunov central limit theorem Corollary
• Polynomial and exponential minimax estimation rates Corollary
• Positive-degree Dolbeault vanishing on pseudoconvex domains Corollary
• The exponential is a continuous bijection from ℝ onto (0,∞) Corollary
• A boundary mode has a different concentration rate Counterexample
• A bounded continuous normalized function that is not of positive type Counterexample
• A discontinuous positive solution of F(x+y)=F(x)F(y) Counterexample
• A generalized Bayes credible set can have zero coverage Counterexample
• A nonintegrable error envelope destroys the linear Hellinger rate Counterexample
• A singular interior mode need not have Gaussian scaling 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
• Omitting the second-stage correction destroys cubic local accuracy Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• The exponential is not uniformly continuous on ℝ Counterexample
• Two closed convex sets can have no strong separator Counterexample
• Diagonal Dirichlet sine-mode heat model with constant forcing Definition
• Generalized Bayes loss update and temperature Definition
• Kullback–Leibler divergence Definition
• Misfit evidence and the potential posterior law Definition
• Sub-Gaussian variables and vectors, and sub-exponential variables Definition
• The Jacobi theta function θ(t)=∑_n∈ℤ e^-π n² t for t>0 Definition
• The six hyperbolic functions and their natural domains Definition
• A Dirac mass has precisely sufficiently negative Sobolev order Example
• A family containing K₁ is viral for vacuous reasons Example
• A finite identifiable model with posterior consistency Example
• A finite-mean polynomial-tail reward is not sub-Gaussian Example
• A Gaussian harmonic leapfrog resonance Example
• A globally Lipschitz Hamiltonian force Example
• A mixture of straight Gaussian paths has a nonlinear marginal velocity Example
• A parameter ledger for the high-girth, high-chromatic alteration proof Example
• A positive mode multiplier too small to certify Example
• A positive non-log-convex solution of the Gamma functional equation Example
• A stable coupled model has an unstable explicit fast microstep Example
• A stiff penalty equation defeats a fixed explicit step Example
• A strictly plurisubharmonic exhaustion of the convex unit ball Example

…and 143 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)