Let f:E→Rf:E\to\mathbb{R}f:E→R (or C\mathbb{C}C) where E⊆RE\subseteq\mathbb{R}E⊆R, and let a∈Ea\in Ea∈E be a limit point of EEE. The function fff is differentiable at aaa 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 (in R\mathbb{R}R or C\mathbb{C}C). This limit (the difference quotient ), when it exists, is the derivative f′(a)f'(a)f′(a). ...

Let U⊆RnU\subseteq\mathbb{R}^nU⊆Rn be open , let f:U→Rmf:U\to\mathbb{R}^mf:U→Rm, and fix a∈Ua\in Ua∈U. Proposition (remainder estimate form of differentiability): The following are equivalent: fff is differentiable at aaa. There exists a linear map A:Rn→RmA:\mathbb{R}^n\to\mathbb{R}^mA:Rn→Rm such that lim⁡h→0∥f(a+h)−f(a)−Ah∥∥h∥=0. \lim_{h\to 0}\frac{\|f(a+h)-f(a)-Ah\|}{\|h\|}=0. limh→0​∥h∥∥f(a+h)−f(a)−Ah∥​=0. Equivalently: there exists a linear map AAA such that for every ε>0\varepsilon>0ε>0 there exists δ>0\delta>0δ>0 with 0<∥h∥<δ ⟹ ∥f(a+h)−f(a)−Ah∥≤ε ∥h∥. 0<\|h\|<\delta \implies \|f(a+h)-f(a)-Ah\|\le \varepsilon\,\|h\|. 0<∥h∥<δ⟹∥f(a+h)−f(a)−Ah∥≤ε∥h∥. In this formulation, AhAhAh is the best linear approximation to fff near aaa and the remainder is “small compared to ∥h∥\|h\|∥h∥.” ...

Differentiability implies continuity: Let U⊆RnU\subseteq\mathbb{R}^nU⊆Rn be open and let f:U→Rmf:U\to\mathbb{R}^mf:U→Rm be differentiable at a∈Ua\in Ua∈U. Then fff is continuous at aaa. This is a basic but essential fact: differentiability is a stronger local property than continuity, and many arguments implicitly use it. Proof sketch: Differentiability at aaa means there is a linear map A:Rn→RmA:\mathbb{R}^n\to\mathbb{R}^mA:Rn→Rm such that lim⁡h→0∥f(a+h)−f(a)−Ah∥∥h∥=0. \lim_{h\to 0}\frac{\|f(a+h)-f(a)-Ah\|}{\|h\|}=0. limh→0​∥h∥∥f(a+h)−f(a)−Ah∥​=0. Hence f(a+h)−f(a)=Ah+o(∥h∥)f(a+h)-f(a)=Ah+o(\|h\|)f(a+h)−f(a)=Ah+o(∥h∥). Taking norms and letting h→0h\to 0h→0 gives ∥f(a+h)−f(a)∥→0\|f(a+h)-f(a)\|\to 0∥f(a+h)−f(a)∥→0, which is continuity at aaa. ...

Let I⊆RI\subseteq\mathbb{R}I⊆R be an interval and let f:I→Rf:I\to\mathbb{R}f:I→R (or C\mathbb{C}C). The function fff is differentiable on III if fff is differentiable at every a∈Ia\in Ia∈I, i.e. if f′(a)f'(a)f′(a) exists for all a∈Ia\in Ia∈I. One often distinguishes interior differentiability and endpoint behavior: on a closed interval [a,b][a,b][a,b], “differentiable on [a,b][a,b][a,b]” may mean differentiable on (a,b)(a,b)(a,b) with one-sided derivatives at aaa and bbb. ...

Let U⊆RkU\subseteq\mathbb{R}^kU⊆Rk be open and let f:U→Rmf:U\to\mathbb{R}^mf:U→Rm. The map fff is differentiable on UUU if it is differentiable at every point a∈Ua\in Ua∈U in the Fréchet sense , i.e. if for each aaa there exists a linear map Df(a):Rk→RmDf(a):\mathbb{R}^k\to\mathbb{R}^mDf(a):Rk→Rm such that ...

Let I⊆RI\subseteq\mathbb{R}I⊆R be an interval and let f,g:I→Rf,g:I\to\mathbb{R}f,g:I→R be differentiable at a point x∈I∘x\in I^\circx∈I∘ (interior of III). Let c∈Rc\in\mathbb{R}c∈R. Proposition (standard derivative rules): Linearity: (f+g)′(x)=f′(x)+g′(x),(cf)′(x)=cf′(x). (f+g)'(x)=f'(x)+g'(x),\qquad (cf)'(x)=c f'(x). (f+g)′(x)=f′(x)+g′(x),(cf)′(x)=cf′(x). Product rule: (fg)′(x)=f′(x)g(x)+f(x)g′(x). (fg)'(x)=f'(x)g(x)+f(x)g'(x). (fg)′(x)=f′(x)g(x)+f(x)g′(x). Quotient rule: if g(x)≠0g(x)\neq 0g(x)=0, then (fg)′(x)=f′(x)g(x)−f(x)g′(x)g(x)2. \left(\frac{f}{g}\right)'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}. (gf​)′(x)=g(x)2f′(x)g(x)−f(x)g′(x)​. Chain rule : if ggg is differentiable at xxx and fff is differentiable at g(x)g(x)g(x), then (f∘g)′(x)=f′(g(x)) g′(x). (f\circ g)'(x)=f'(g(x))\,g'(x). (f∘g)′(x)=f′(g(x))g′(x). These formulas are the computational backbone of differential calculus; they are proved directly from the limit definition of the derivative . ...

Dini’s Theorem: Let KKK be a compact metric space and let fn:K→Rf_n:K\to\mathbb{R}fn​:K→R be continuous for all nnn. Suppose: for each x∈Kx\in Kx∈K, the sequence fn(x)f_n(x)fn​(x) is monotone in nnn (either nondecreasing for all xxx, or nonincreasing for all xxx), and fn(x)→f(x)f_n(x)\to f(x)fn​(x)→f(x) pointwise on KKK for some continuous function f:K→Rf:K\to\mathbb{R}f:K→R. Then fn→ff_n\to ffn​→f uniformly on KKK. ...

Let U⊆RkU\subseteq\mathbb{R}^kU⊆Rk be open , let f:U→Rmf:U\to\mathbb{R}^mf:U→Rm, let a∈Ua\in Ua∈U, and let v∈Rkv\in\mathbb{R}^kv∈Rk. The directional derivative of fff at aaa in the direction vvv is Dvf(a):=HAHAHUGOSHORTCODE619s1HBHBh→0f(a+hv)−f(a)h, D_v f(a) := lim _{h\to 0}\frac{f(a+hv)-f(a)}{h}, Dv​f(a):=HAHAHUGOSHORTCODE619s1HBHBh→0​hf(a+hv)−f(a)​, provided the limit exists in Rm\mathbb{R}^mRm. ...

Dirichlet Test: Let (an)(a_n)(an​) and (bn)(b_n)(bn​) be real sequences such that: the partial sums AN=∑n=1NanA_N=\sum_{n=1}^N a_nAN​=∑n=1N​an​ are bounded, and bnb_nbn​ is monotone with bn→0b_n\to 0bn​→0. Then the series ∑n=1∞anbn\sum_{n=1}^\infty a_n b_n∑n=1∞​an​bn​ converges . Dirichlet’s test is a powerful tool for oscillatory series where cancellation occurs through bounded partial sums. ...

Let XXX be a set . A metric (or distance function) on XXX is a function d:X×X→[0,∞)d:X\times X\to[0,\infty)d:X×X→[0,∞) such that for all x,y,z∈Xx,y,z\in Xx,y,z∈X: (Positive definiteness) d(x,y)=0d(x,y)=0d(x,y)=0 iff x=yx=yx=y. (Symmetry) d(x,y)=d(y,x)d(x,y)=d(y,x)d(x,y)=d(y,x). (Triangle inequality) d(x,z)≤d(x,y)+d(y,z)d(x,z)\le d(x,y)+d(y,z)d(x,z)≤d(x,y)+d(y,z). Metrics quantify “closeness” abstractly. Most of analysis can be formulated in terms of a metric, including convergence , continuity , compactness , and completeness . ...