Foundations

A finite limit controls output error through sufficiently small input distance.

A derivative is a finite limit of local difference quotients.

A definite integral is the limiting signed accumulation of small contributions.

Continuity connects differentiation with integration.

Real convergence can be checked by distances between late terms.

A series converges through its partial sums.

The remainder quantifies a finite polynomial approximation.

One cutoff controls the error at every domain point.

Integration starts from measurable simple functions.

Differentiability means a linear approximation with smaller-order error.

A continuous double integral can be evaluated by slicing.

These coordinate derivatives have different input and output types.

Circulation around a boundary equals oriented curl flux.

Outward boundary flux equals integrated divergence.

A continuous real function on a closed bounded interval attains a maximum and minimum and takes every value between its endpoint values.

Differentiable real functions satisfy (fg)′=f′g+fg′ and (g∘f)′=(g′∘f)f′.

If f is continuous on [a,b] and differentiable on (a,b), a<b, some c∈(a,b) satisfies f′(c)=(f(b)−f(a))/(b−a).

For f differentiable throughout an open real interval, its derivative takes every value strictly between f′(a) and f′(b) at some c∈(a,b), whenever a<b are points of that interval.

For g of class C¹ on [a,b] and f continuous on an interval containing g([a,b]), ∫ₐᵇ f(g(x))g′(x)dx=∫g(a)ᵍ(b)f(u)du.

For u,v of class C¹ on [a,b], ∫ₐᵇuv′dx=[uv]ₐᵇ−∫ₐᵇu′v dx.

For bounded real f on [a,b], lower and upper Darboux sums use each subinterval's infimum and supremum.

A bounded real function on [a,b] is Riemann integrable exactly when its discontinuity set has Lebesgue measure zero.

If real functions fₙ on one set A satisfy |fₙ(x)|≤Mₙ for all x∈A, with Mₙ≥0 and ΣMₙ<∞, then Σfₙ converges uniformly and absolutely on A.

An improper integral is the finite limit of proper integrals as a singular endpoint or an infinite endpoint is approached.

For positive continuous decreasing f on [1,∞), Σₙ₌₁∞f(n) and ∫₁∞f(x)dx converge or diverge together.

If bₙ≥0 decreases to zero, Σ(−1)ⁿbₙ converges; the absolute error after N terms is at most the next omitted magnitude b_(N+1).

For increasing nonnegative measurable fₙ→f, ∫fₙdμ→∫f dμ.

If measurable real fₙ→f pointwise and |fₙ|≤g for one integrable g, then ∫fₙdμ→∫f dμ.

A function is holomorphic on an open set when it is complex differentiable at every point.

For a piecewise C¹ path γ, ∫γf(z)dz=∫f(γ(t))γ′(t)dt.