Density of Q\mathbb{Q}Q in R\mathbb{R}R: If a<ba<ba<b are real numbers, then there exists q∈Qq\in\mathbb{Q}q∈Q such that a<q<b.a<q<b.a<q<b. This ensures rationals approximate reals arbitrarily well and is foundational for approximation arguments, constructions via sequences, and separating points in analysis. Proof sketch (optional): Choose n∈Nn\in\mathbb{N}n∈N with n(b−a)>1n(b-a)>1n(b−a)>1 (Archimedean property ). Then pick an integer mmm with na<m<nbna<m<nbna<m<nb (using existence of integers between reals). Set q=m/nq=m/nq=m/n. ...

Density of R∖Q\mathbb{R}\setminus\mathbb{Q}R∖Q in R\mathbb{R}R: If a<ba<ba<b are real numbers, then there exists an irrational number x∈R∖Qx\in\mathbb{R}\setminus\mathbb{Q}x∈R∖Q such that a<x<b.a<x<b.a<x<b. This shows that irrationals are just as ubiquitous as rationals and prevents “gaps” of irrationality; it is often used in constructing sequences with prescribed properties. Proof sketch (optional): By density of ℚ , choose q∈Qq\in\mathbb{Q}q∈Q with a<q<ba<q<ba<q<b. Then choose a small rational r>0r>0r>0 so that q+r2∈(a,b)q+r\sqrt{2}\in(a,b)q+r2​∈(a,b) (possible since r2r\sqrt{2}r2​ can be made arbitrarily small). The number q+r2q+r\sqrt{2}q+r2​ is irrational. ...

Let f:E→Rf:E\to\mathbb{R}f:E→R (or C\mathbb{C}C) with E⊆RE\subseteq\mathbb{R}E⊆R and let a∈Ea\in Ea∈E be a limit point of EEE. If the limit lim⁡x→af(x)−f(a)x−a\lim_{x\to a}\frac{f(x)-f(a)}{x-a}x→alim​x−af(x)−f(a)​ exists (this is the difference quotient ), it is called the derivative of fff at aaa and is denoted f′(a)f'(a)f′(a). ...

Derivative sign implies monotonicity: Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be continuous on [a,b][a,b][a,b] and differentiable on (a,b)(a,b)(a,b). If f′(x)≥0f'(x)\ge 0f′(x)≥0 for all x∈(a,b)x\in(a,b)x∈(a,b), then fff is nondecreasing on [a,b][a,b][a,b] (i.e., x<yx<yx<y implies f(x)≤f(y)f(x)\le f(y)f(x)≤f(y)). If f′(x)≤0f'(x)\le 0f′(x)≤0 for all x∈(a,b)x\in(a,b)x∈(a,b), then fff is nonincreasing on [a,b][a,b][a,b]. If f′(x)>0f'(x)>0f′(x)>0 for all x∈(a,b)x\in(a,b)x∈(a,b), then fff is strictly increasing on [a,b][a,b][a,b] (and similarly f′<0f'<0f′<0 implies strictly decreasing). This is one of the most direct ways analysis turns differential information into global order information, and it is frequently used to prove injectivity and existence of inverses . ...

Let I⊆RI\subseteq\mathbb{R}I⊆R be an interval and let f:I→Rf:I\to\mathbb{R}f:I→R be differentiable on I∘I^\circI∘. Proposition: If f′(x)=0f'(x)=0f′(x)=0 for all x∈I∘x\in I^\circx∈I∘, then fff is constant on III; i.e., for all x,y∈Ix,y\in Ix,y∈I one has f(x)=f(y)f(x)=f(y)f(x)=f(y). This is a fundamental rigidity result: having zero instantaneous rate of change everywhere forces no global change. ...

Let (X,d)(X,d)(X,d) be a metric space and let A⊆XA\subseteq XA⊆X. The derived set of AAA, denoted A′A'A′, is the set of all limit points of AAA: A′:={x∈X:x is a limit point of A}.A' := \{x\in X : x\ \text{is a limit point of }A\}.A′:={x∈X:x is a limit point of A}.The derived set isolates where a set “accumulates.” It is useful in studying closed sets, perfect sets, and in iterative constructions like the Cantor–Bendixson process. ...

Let A:Rn→RnA:\mathbb{R}^n\to\mathbb{R}^nA:Rn→Rn be a linear map . Saying det⁡A≠0\det A\neq 0detA=0 is equivalent to saying AAA is invertible. Stability of invertibility (Neumann series lemma): If AAA is invertible and BBB is another linear map such that ∥A−1(B−A)∥<1, \|A^{-1}(B-A)\|<1, ∥A−1(B−A)∥<1, then BBB is invertible and B−1=∑k=0∞(−A−1(B−A))k A−1. B^{-1}=\sum_{k=0}^\infty \bigl(-A^{-1}(B-A)\bigr)^k\,A^{-1}. B−1=∑k=0∞​(−A−1(B−A))kA−1. Moreover, ∥B−1∥≤∥A−1∥1−∥A−1(B−A)∥. \|B^{-1}\|\le \frac{\|A^{-1}\|}{1-\|A^{-1}(B-A)\|}. ∥B−1∥≤1−∥A−1(B−A)∥∥A−1∥​. In particular, if ∥B−A∥≤12∥A−1∥\|B-A\|\le \frac{1}{2\|A^{-1}\|}∥B−A∥≤2∥A−1∥1​ then BBB is invertible and ∥B−1∥≤2∥A−1∥\|B^{-1}\|\le 2\|A^{-1}\|∥B−1∥≤2∥A−1∥. ...

Let (X,d)(X,d)(X,d) be a metric space and let A⊆XA\subseteq XA⊆X. The diameter of AAA is diam⁡(A):=sup⁡{d(x,y):x,y∈A}∈[0,∞].\operatorname{diam}(A):=\sup\{d(x,y): x,y\in A\}\in[0,\infty].diam(A):=sup{d(x,y):x,y∈A}∈[0,∞]. (If the set of distances is unbounded, the supremum is +∞+\infty+∞.) Diameter measures the “size” of a set in terms of its maximal pairwise separation. It is used in compactness and completeness arguments (e.g., nested closed sets with diameters →0\to 0→0). ...

A diffeomorphism between open sets U,V⊆RnU,V \subseteq \mathbb{R}^nU,V⊆Rn is a map f:U→Vf:U\to Vf:U→V such that fff is bijective , fff is continuously differentiable on UUU (i.e., f∈C1(U,V)f\in C^1(U,V)f∈C1(U,V)), and the inverse map f−1:V→Uf^{-1}:V\to Uf−1:V→U is also continuously differentiable (i.e., f−1∈C1(V,U)f^{-1}\in C^1(V,U)f−1∈C1(V,U)). Diffeomorphisms are the “smooth isomorphisms” of Euclidean spaces: they preserve the differentiable structure and are the natural maps appearing in the inverse function theorem and change-of-variables formula . ...

Let f:E→Rf:E\to\mathbb{R}f:E→R (or C\mathbb{C}C) with E⊆RE\subseteq\mathbb{R}E⊆R. For a∈Ea\in Ea∈E and x∈Ex\in Ex∈E with x≠ax\ne ax=a, the difference quotient of fff at (a,x)(a,x)(a,x) is f(x)−f(a)x−a.\frac{f(x)-f(a)}{x-a}.x−af(x)−f(a)​.Difference quotients are the finite approximations to the derivative. The derivative f′(a)f'(a)f′(a), when it exists, is the limit of these quotients as x→ax\to ax→a. ...