Right derivative and left derivative

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 E∩(a,∞)E\cap(a,\infty)E∩(a,∞) and of E∩(−∞,a)E\cap(-\infty,a)E∩(−∞,a). The right derivative of fff at aaa is f+′(a):=lim⁡h↓0f(a+h)−f(a)h,f'_+(a):=\lim_{h\downarrow 0}\frac{f(a+h)-f(a)}{h},f+′(a):=h↓0limhf(a+h)−f(a), provided the limit exists. The left derivative is ...

Rolle's Theorem

Rolle’s Theorem: 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(a)=f(b)f(a)=f(b)f(a)=f(b), then there exists c∈(a,b)c\in(a,b)c∈(a,b) such that f′(c)=0.f'(c)=0.f′(c)=0. Rolle’s theorem is the key step in proving the mean value theorem and links global behavior (endpoint values) to local behavior (vanishing derivative ). ...

Root Test

Root Test: For a series ∑an\sum a_n∑an (real or complex), define α=lim sup⁡n→∞∣an∣n.\alpha=\limsup_{n\to\infty}\sqrt[n]{|a_n|}.α=limsupn→∞n∣an∣. If α<1\alpha<1α<1, then ∑an\sum a_n∑an converges absolutely . If α>1\alpha>1α>1 (or α=∞\alpha=\inftyα=∞), then ∑an\sum a_n∑an diverges . If α=1\alpha=1α=1, the test is inconclusive. The root test is well-suited for expressions like ∣an∣=(something)n|a_n|=(\text{something})^n∣an∣=(something)n. ...

Schwarz's Theorem (Clairaut's theorem)

Schwarz (Clairaut) Theorem: Let U⊆RnU\subseteq\mathbb{R}^nU⊆Rn be open and let f:U→Rf:U\to\mathbb{R}f:U→R. Fix indices i≠ji\neq ji=j. If the mixed second partial derivatives ∂2f∂xi∂xj\frac{\partial^2 f}{\partial x_i\partial x_j}∂xi∂xj∂2f and ∂2f∂xj∂xi\frac{\partial^2 f}{\partial x_j\partial x_i}∂xj∂xi∂2f exist on a neighborhood of a∈Ua\in Ua∈U and are continuous at aaa, then ∂2f∂xi∂xj(a)=∂2f∂xj∂xi(a). \frac{\partial^2 f}{\partial x_i\partial x_j}(a)=\frac{\partial^2 f}{\partial x_j\partial x_i}(a). ∂xi∂xj∂2f(a)=∂xj∂xi∂2f(a). ...

Second derivative tests

One-variable second derivative test Let f:I→Rf:I\to\mathbb{R}f:I→R be twice differentiable and let a∈I∘a\in I^\circa∈I∘ satisfy f′(a)=0f'(a)=0f′(a)=0. Proposition (one variable): If f′′(a)>0f''(a)>0f′′(a)>0, then aaa is a strict local minimum of fff. If f′′(a)<0f''(a)<0f′′(a)<0, then aaa is a strict local maximum of fff. If f′′(a)=0f''(a)=0f′′(a)=0, no conclusion follows in general. Multivariable Hessian test Let U⊆RnU\subseteq\mathbb{R}^nU⊆Rn be open and let f:U→Rf:U\to\mathbb{R}f:U→R be $C^2$ . Let a∈Ua\in Ua∈U be a critical point , i.e. HAHAHUGOSHORTCODE779s5HBHB=0$\\nabla f(a)$ =0HAHAHUGOSHORTCODE779s5HBHB=0, and let H=HAHAHUGOSHORTCODE779s6HBHBH=$\\nabla^2 f(a)$ H=HAHAHUGOSHORTCODE779s6HBHB be the Hessian matrix at aaa. ...

Separated sets

Let (X,d)(X,d)(X,d) be a metric space and let A,B⊆XA,B\subseteq XA,B⊆X. The sets AAA and BBB are separated if A‾∩B=∅andA∩B‾=∅.\overline{A}\cap B=\varnothing \quad\text{and}\quad A\cap \overline{B}=\varnothing.A∩B=∅andA∩B=∅.Separatedness is a symmetric condition stronger than disjointness: it requires each set to have a neighborhood that misses the other in a closure sense. Separations are a standard way to characterize disconnectedness. ...

Sequential characterization of closed sets

Sequential characterization of closed sets: Let (X,d)(X,d)(X,d) be a metric space and F⊆XF\subseteq XF⊆X. Then FFF is closed if and only if whenever (xn)(x_n)(xn) is a sequence in FFF with xn→xx_n\to xxn→x in XXX, one has x∈Fx\in Fx∈F. This gives a practical criterion for closedness using sequences, avoiding direct work with complements or open balls . ...

Sequential characterization of closure

Sequential characterization of closure: Let (X,d)(X,d)(X,d) be a metric space and E⊆XE\subseteq XE⊆X. A point x∈Xx\in Xx∈X belongs to the closure E‾\overline{E}E if and only if there exists a sequence (xn)(x_n)(xn) in EEE such that xn→x.x_n\to x.xn→x. This result ties topological notions (closure) to analytic ones (sequences) and is one reason sequences are so effective in metric spaces. ...

Sequential compactness equals compactness (metric spaces)

Sequential compactness equals compactness: Let (X,d)(X,d)(X,d) be a metric space and K⊆XK\subseteq XK⊆X. Then KKK is compact (every open cover has a finite subcover) if and only if KKK is sequentially compact (every sequence in KKK has a convergent subsequence with limit in KKK). This equivalence is special to metric (first countable) spaces and makes compactness usable via sequences, which is often the most practical viewpoint in analysis. ...

Sequentially compact set

Let (X,d)(X,d)(X,d) be a metric space and let K⊆XK\subseteq XK⊆X. The set KKK is sequentially compact if for every sequence (xn)(x_n)(xn) in KKK there exist: a subsequence (xnk)(x_{n_k})(xnk), and a point x∈Kx\in Kx∈K such that xnk→xas k→∞x_{n_k}\to x \quad\text{as }k\to\inftyxnk→xas k→∞ (see convergence ). In metric spaces, sequential compactness is equivalent to compactness , but the sequential formulation is often easier to use in analysis. ...