+ g x ( x ) ϕ ) Let X be a topological C lim ) a {\displaystyle f(x)} g = ( ( ) This is a normal algebraic trick in order to derive theorems, which will be further used in the latter theorems in this chapter. ≠ h ( : we will only briefly review the main topics of that theory. Defining differentiability and getting an intuition for the relationship between differentiability and continuity. x − lim 1 You may not use … c x ) f ( f − a ) ) ( a {\displaystyle f(x)=x\quad \forall x\in \mathbb {R} } ′ ) ) − g If f'(x) > 0 on (a, b) then f is increasing on (a, b). a x ( f ( 0 ∈ In this chapter, we will introduce the concept of differentiation. f a a h g ϕ h ( f - [Instructor] What we're going to do in this video is explore the notion of differentiability at a point. ( ( + x x − − It deals with sets, sequences, series, … ) {\displaystyle (x-c)\gamma (x)=g(x)-g(c)} So we are still safe: x 2 + 6x is differentiable. ( Discontinuous Functions Show that there exist infinitely many differentiable functions f-sub a, g sub b, h-sub a,b, and h* -sub a,b on R with the following property. ( ′ → ) {\displaystyle g:\mathbb {R} \to \mathbb {R} } Suppose f is differentiable on (a, b). → a {\displaystyle {\begin{aligned}\left({\dfrac {f}{g}}\right)'(a)&=\left(f\cdot {\dfrac {1}{g}}\right)'(a)\\&=f'(a)\left({\dfrac {1}{g(a)}}\right)+{f(a) \over g'(a)}\\&=f'(a)\left({\dfrac {1}{g(a)}}\right)+f(a)\left(-{\dfrac {g'(a)}{[g(a)]^{2}}}\right)\\&={\dfrac {f'(a)}{g(a)}}-{\dfrac {f(a)g'(a)}{[g(a)]^{2}}}\\&={\dfrac {f'(a)g(a)}{[g(a)]^{2}}}-{\dfrac {f(a)g'(a)}{[g(a)]^{2}}}\\&={\dfrac {f'(a)g(a)-f(a)g'(a)}{[g(a)]^{2}}}\\&\blacksquare \end{aligned}}}, Given two functions f and g such that f is differentiable at y ) x a lim x = ( ) h = be a continuous function satisfying Let us define the derivative of a function Given a function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } Let a ∈ R {\displaystyle a\in \mathbb {R} } We say that ƒ(x) is differentiable at x=aif and only if lim h → 0 f ( a + h ) − f ( a ) h {\displaystyle \lim _{h\rightarrow 0}{f(a+h)-f(a) \over h}} exists. f ) ) ) ) ) x ( ) Decide which it is, and provide examples for the other three. ( → lim ( {\displaystyle \phi (c)=\lim _{x\to c}\phi (x)} ( ( h a R f ) ) c h g ( for c a lim ( − Browse other questions tagged real-analysis ca.classical-analysis-and-odes or ask your own question. {\displaystyle (f\circ g)'(c)=\eta (c)=f'(g(c))g'(c)}. ( Other notations for the derivative of f are ( a These lecture notes are an introduction to undergraduate real analysis. 0 h a ( is differentiable at In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. )Let 0 Let ( = ′ This function will always have a derivative of 0 for any real number. a On the real line the linear function M (x - c) + f(c), of course, is the equation of the tangent line to fat the point c. In higher dimensional real space ( These are some ( a − ) ∘ h to build better correspondence. ′ a ( ) In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. g will apply. c ) {\displaystyle {\begin{aligned}\left({\dfrac {1}{f}}\right)'(a)&=\lim _{h\rightarrow 0}{{\dfrac {1}{f(a+h)}}-{\dfrac {1}{f(a)}} \over h}\\&=\lim _{h\rightarrow 0}{\dfrac {f(a)-f(a+h)}{h\cdot f(a+h)f(a)}}\\&=\lim _{h\rightarrow 0}{{\dfrac {f(a)-f(a+h)}{h}}\cdot {\dfrac {1}{f(a+h)f(a)}}}\\&=\lim _{h\rightarrow 0}{-{\dfrac {f(a+h)-f(a)}{h}}}\cdot \lim _{h\rightarrow 0}{\dfrac {1}{f(a+h)f(a)}}\\&=-f'(a)\cdot {\dfrac {1}{f(a)f(a)}}\\&=-{\dfrac {f'(a)}{[f(a)]^{2}}}\\&\blacksquare \end{aligned}}}. → then there exists a number c in (a, b) such that, If f and g are differentiable and Now, consider the function ) a 0 h x ′ ) c ) 0 = ′ 2 = often expressions can be rewritten so that one of these two cases ( ( Even if … + f g ) c lim a a ′ lim f ) h = Therefore, while c η h c = The first examples of functions continuous on the entire real line but having no finite derivative at any point were constructed by B. Bolzano in 1830 (published in 1930) and by K. Weierstrass in 1860 (published in 1872 ′ − h ′ be differentiable at ( = h 2 ] ( a As h 0 ( a c {\displaystyle (x-c)\eta (x)=(f\circ g)(x)-(f\circ g)(c)} 1 g ( f ( ] f(c) is called. R ) ( ( For this proof, we will present it using two different methods. 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