Fubini’s Theorem (Riemann, continuous case): Let f:[a,b]×[c,d]→Rf:[a,b]\times[c,d]\to\mathbb{R}f:[a,b]×[c,d]→R be continuous . Define g(x)=∫cdf(x,y) dy(x∈[a,b]),h(y)=∫abf(x,y) dx(y∈[c,d]). g(x)=\int_c^d f(x,y)\,dy \quad (x\in[a,b]), \qquad h(y)=\int_a^b f(x,y)\,dx \quad (y\in[c,d]). g(x)=∫cd​f(x,y)dy(x∈[a,b]),h(y)=∫ab​f(x,y)dx(y∈[c,d]). Then ggg and hhh are continuous, and the double integral exists and satisfies $ \int_a^b\left(\int_c^d f(x,y),dy\right)dx \int_{[a,b]\times[c,d]} f(x,y),d(x,y) \int_c^d\left(\int_a^b f(x,y),dx\right)dy. $ ...

A function (or map) from a set XXX to a set YYY is a rule that assigns to each x∈Xx\in Xx∈X exactly one element of YYY, denoted f(x)f(x)f(x). Formally, it is a subset f⊆X×Yf\subseteq X\times Yf⊆X×Y (a Cartesian product ) such that: for every x∈Xx\in Xx∈X there exists y∈Yy\in Yy∈Y with (x,y)∈f(x,y)\in f(x,y)∈f, and if (x,y1)∈f(x,y_1)\in f(x,y1​)∈f and (x,y2)∈f(x,y_2)\in f(x,y2​)∈f, then y1=y2y_1=y_2y1​=y2​. One writes f:X→Yf:X\to Yf:X→Y and calls XXX the domain and YYY the codomain . Functions are the primary objects of study in analysis: limits, continuity , differentiability , and integrability are properties of functions. ...

Fundamental Theorem of Calculus (Part I): Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be Riemann integrable and define F(x)=∫axf(t) dt(x∈[a,b]). F(x)=\int_a^x f(t)\,dt \qquad (x\in[a,b]). F(x)=∫ax​f(t)dt(x∈[a,b]). Then FFF is continuous on [a,b][a,b][a,b]. Moreover, if fff is continuous at a point x0∈(a,b)x_0\in(a,b)x0​∈(a,b), then FFF is differentiable at x0x_0x0​ and F′(x0)=f(x0). F'(x_0)=f(x_0). F′(x0​)=f(x0​). ...

Fundamental Theorem of Calculus (Part II): Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be continuous . Suppose F:[a,b]→RF:[a,b]\to\mathbb{R}F:[a,b]→R is differentiable on (a,b)(a,b)(a,b), continuous on [a,b][a,b][a,b], and satisfies F′(x)=f(x)for all x∈(a,b). F'(x)=f(x)\quad\text{for all }x\in(a,b). F′(x)=f(x)for all x∈(a,b). Then ∫abf(x) dx=F(b)−F(a). \int_a^b f(x)\,dx = F(b)-F(a). ∫ab​f(x)dx=F(b)−F(a). ...

Let f:E→Rf:E\to\mathbb{R}f:E→R and let a∈Ea\in Ea∈E. The point aaa is a global maximum (or absolute maximum) of fff on EEE if ∀x∈E, f(x)≤f(a).\forall x\in E,\ f(x)\le f(a).∀x∈E, f(x)≤f(a). The point aaa is a global minimum (or absolute minimum) of fff on EEE if ∀x∈E, f(a)≤f(x).\forall x\in E,\ f(a)\le f(x).∀x∈E, f(a)≤f(x). Global extrema are stronger than local extrema and need not exist in general. A central theorem in analysis is that continuous functions on compact sets attain both a global maximum and a global minimum. ...

Let U⊆RkU\subseteq\mathbb{R}^kU⊆Rk be open and let f:U→Rf:U\to\mathbb{R}f:U→R be differentiable (at least with existing partial derivatives ) at a∈Ua\in Ua∈U. The gradient of fff at aaa is the vector ∇f(a):=(∂f∂x1(a),…,∂f∂xk(a))∈Rk. \nabla f(a) := \left(\frac{\partial f}{\partial x_1}(a),\dots,\frac{\partial f}{\partial x_k}(a)\right)\in\mathbb{R}^k. ∇f(a):=(∂x1​∂f​(a),…,∂xk​∂f​(a))∈Rk.When fff is differentiable at aaa, ∇f(a)\nabla f(a)∇f(a) represents the linear functional “best approximating” changes in fff near aaa, via ...

Greatest Lower Bound Theorem: If E⊆RE\subseteq \mathbb{R}E⊆R is nonempty and bounded below , then inf⁡E\inf EinfE exists in R\mathbb{R}R. This is the “lower” counterpart to the least upper bound theorem and follows immediately by applying the supremum property to −E={−x:x∈E}-E=\{-x:x\in E\}−E={−x:x∈E}. Proof sketch (optional): If EEE is bounded below, then −E-E−E is bounded above. Let s=sup⁡(−E)s=\sup(-E)s=sup(−E). Then −s-s−s is the greatest lower bound of EEE, i.e., inf⁡E=−s\inf E=-sinfE=−s. ...

Heine–Borel Theorem: A subset K⊆RkK\subseteq \mathbb{R}^kK⊆Rk is compact (in the Euclidean metric) if and only if it is closed and bounded . This theorem is the fundamental compactness criterion in Euclidean spaces and is used constantly to verify hypotheses of the extreme value theorem , uniform continuity , and convergence results. Proof sketch (optional): If KKK is compact, it is bounded (cover by balls of radius 1 and extract a finite subcover) and closed (limits of sequences in KKK stay in KKK). Conversely, if KKK is closed and bounded, any sequence in KKK is bounded, hence has a convergent subsequence by Bolzano–Weierstrass ; closedness ensures the limit lies in KKK, giving sequential compactness and hence compactness. ...

Heine–Cantor Theorem: Let (X,dX)(X,d_X)(X,dX​) be a compact metric space and let (Y,dY)(Y,d_Y)(Y,dY​) be a metric space. If f:X→Yf:X\to Yf:X→Y is continuous , then fff is uniformly continuous on XXX; i.e., ∀ε>0 ∃δ>0 ∀x,y∈X: dX(x,y)<δ⇒dY(f(x),f(y))<ε.\forall \varepsilon>0\;\exists \delta>0\;\forall x,y\in X:\ d_X(x,y)<\delta \Rightarrow d_Y(f(x),f(y))<\varepsilon.∀ε>0∃δ>0∀x,y∈X: dX​(x,y)<δ⇒dY​(f(x),f(y))<ε. ...

Let U⊆RkU\subseteq\mathbb{R}^kU⊆Rk be open and let f:U→Rf:U\to\mathbb{R}f:U→R. If all second partial derivatives exist at a∈Ua\in Ua∈U, the Hessian matrix of fff at aaa is the k×kk\times kk×k matrix Hf(a):=[∂2f∂xi ∂xj(a)]1≤i,j≤k. H_f(a) := \left[\frac{\partial^2 f}{\partial x_i\,\partial x_j}(a)\right]_{1\le i,j\le k}. Hf​(a):=[∂xi​∂xj​∂2f​(a)]1≤i,j≤k​.When fff is $C^2$ , the Hessian is symmetric (mixed partials commute) and governs second-order Taylor approximation and second-derivative tests for extrema. ...