Differentiability Definition and 197 Threads

given a function F:S-->R such that for every element belonging to "S" has both left hand derivative and right hand derivative and are equal to the derivative at that point. Can we say that the function is differentiable..?

Homework Statement Can someone tell me why and why not following functions are Continous and Differentiable. I am also providing the answer but can some help me understand.. thanks 1) f(x) = x^(2/3) -1 on [-8,8] answer: function is continuous but not differentiable on -8. Is that...

Here's a question I've thought about on several occasions: How many levels of derivatives (rates of change) typically occur for objects in nature? For instance, a car has a position, velocity (1st derivative), and acceleration (2nd derivative), but it can also be said to have a rate of...

[b]1. Suppose that f is differentiable on R and \lim_{x \rightarrow \infty} f'(x) = M. Show that \lim_{x \rightarrow \infty} (f(x+1)-f(x)) also exists, and compute it. [b]3. I am pretty sure the limit will be equal to m. Here is my attempt. \lim_{x \rightarrow \infty} f'(x) = \lim_{x...

This isn't homework per se... It's a question from a book I'm self-studying from. If f is continuous on [a,b] and differentiable at a point c \in [a,b], show that, for some pair m,n \in \mathbb{N}, \left | \frac{f(x)-f(c)}{x-c}\right | \leq n whenever 0 \leq |x-c| \leq \frac{1}{m}...

Ok, so I have f(x,y)=(p(x)+q(y))/(x^2+y^2) where (x,y)NOT=0 and f(0,0)=0. the basic idea of the function is that the numerator contains 2 polynomials>2nd order. and the denominator has a Xsquared+ysquared. I have to prove that if f(x,y) is differentiable at (0,0) then its partial derivatives...

Hi, could somebody please help me with the following question, I have been stuck on it for ages. [b]1. let f[0,1] -> R be continuous with f(0)=0, f(1)=1. Prove the following: a.(i) If for c in (0,1) f is differentiable at c with f'(c)<0 then there are exists points y such that f(x)=y has...

Let K>0 and a>0. The function f is said to satisfy the Lipschitz condition if |f(x)-f(y)|<= K |x-y|a .. I am given a problem where I must prove that f is differentiability if a>1. I know I need to show that limx->c(f(x)-f(c))/ (x-c) exists. I am having quite a hard time. Any hints?

Let f:(a,infinity)-->R (reals) be differentiable and let A and B be real numbers. Prove that if f(x)-->A and f '(x)-->B as x --> infinity, then B=0. I would guess that the mean value theorem may be needed but I am not sure how to use it considering that we're dealing with x --> infinity.

Let f(x)= sinx/x if x \neq 0 and f(0)=1 Find a polynomial pN of degree N so that |f(x)-pN(x)| \leq |x|^(N+1) for all x. Argue that f is differentiable, f' is differentiable, f" is differentiale .. (all derivatives exist at all points). I'm not sure about this one at all. Can you guys...

a. Suppose f is twice differentiable on (0, infinity). Suppose that |f(x)|< or equal A0 for all x>0 and that the second derivative satisfies |f''(x)|< or equal A2 for all x>0. Prove that for all x>0 and all h>0 |f'(x)| < or equal 2A0/h + A2h/2 This is sometimes called Landau's inequality...

Homework Statement y= y= {1+3ax+2x^2} if x is < or = 1 {mx+a} if x>0 what values for m and a make x continuous and differentiable at 1? Homework Equations n/a The Attempt at a Solution i solved for when x=1. i got 3+3a. this is also the right hand...

Homework Statement Find the values of the constants a and b such that the function f(x) is differentiable on R Homework Equations f(x) = ax2 if x < 2 f(x) = -4(x-3) + b if x >= 2 The Attempt at a Solution ax2 = -4(x-3) + b 2xa = -4x...

Hello, I was wondering where I can find a proof to the following theorem: If F is differentiable at every pt. of [a,b] and if F' is lebesque on [a,b] then F(x) = int(f,dt,a,x) a<=x<=b. And the converse. He gives the theorem are page 324 and a reference in his bibliography. I...

Let f:]a,b[\to\mathbb{R} be a differentiable function. For each fixed x\in ]a,b[, we can define a function \epsilon_x: D_x\to\mathbb{R},\quad\quad \epsilon_x(u) = \frac{f(x+u) - f(x)}{u} \;-\; f'(x) where D_x = \{u\in\mathbb{R}\backslash\{0\}\;|\; a < x+u < b\}. Now we have...

Is there a geometric meaning for the derivative of a complex valued function, or any other motivation for the derivative?

Suppose the real valued g is defined on \mathbb{R} and g'(x) < 0 for every real x. Prove there's no differentiable f: R \rightarrow R such that f \circ f = g.

1. The problem statement. Give a complete and accurate \delta - \epsilon proof of the thereom: If f is differentiable at a, then f is continuous at a. 2. The attempt at a solution Known: \forall\epsilon>0, \exists\delta>0, \forall x, |x-a|<\delta \implies \left|\frac{f(x) - f(a)}{x-a}...

Define f(0,0) = 0 and f(x,y) = \frac {x^3}{x^2 + y^2} if (x,y) \neq (0,0) a) Prove that the partial derivatives of f are bounded functions in R^2. b) Let \mathbf{u} be any unit vector in R^2. Show that the directional derivative (D_{\mathbf{u}} f)(0,0) exists, and its absolute value is at...

Hello: These are not h/w problems but something from the class notes which I am not able to fully understand. I have two questions stated below Question1 We are doing chain rule and function of several variables. To explain the prof has first explained about single variables and then...

Homework Statement Let f: R -> R be a continuous function such that f '(x) exists for all x =/= 0 . Say also that the limit of f '(x) as x goes to 0 exists and is equal to L. Must f '(0) exist as well? Prove or disprove. The Attempt at a Solution I can't come up with a proof or...

Hi guys, I have this problem understanding that holomorpic functions must be infinitely differentiable. Indeed, it does follow from the Cauchy formula. But take z=x+iy. It satisfies C-R equations and has a first derivative = 1. I fail to see how this function is infinitely differentiable...

I have two problems. I posted the first problem before but I still can´t solve it. Homework Statement Find and classify the critical points of f(x,y,z) = xy + xz + yz + x^3 + y^3 + z^3 Homework Equations - The Attempt at a Solution df/dx = y + z +3x^2, df/dy = x + z + 3y^2...

Hi. How do I show that f is differentiable, but f' is discontinuous at 0? I guess I'm just looking for a general idea to show discontinuity. Thanks

Homework Statement Homework Equations The Attempt at a Solution I think to determine where it's differentiable it has something to do with partial derivatives. But I am just so completely clueless on how to even start this guy off that any tips or minor suggestions on where to...

Homework Statement Suppose f:(0,\infty)->R and f(x)-f(y)=f(x/y) for all x,y in (0,\infty) and f(1)=0. Prove f is continuous on (0,\infty) iff f is continuous at 1. Homework Equations I think I ought to use these defn's of continuity: f continuous at a iff f(x)->f(a) as x->a or f is cont at a...

I've been thinking... Since derivatives can be written as: f'(c)= \lim_{x\rightarrow{c}}\frac{f(x)-f(c)}{x-c} and for the limit to exist, it's one sided limits must exist also right? So if the one sided limits exist, and thus the limit as x approaches c (therefore the derivative at c)...

Homework Statement Given an interval I. The function f goes from I to the real line. Define f as f(x)=x^2 if x rational or 0 if x is irrational. show that f i differentiablle at x=0 and find its derivative at this point, i.e. x=0. Homework Equations I have a given lead on this. That...

[SOLVED] measure zero and differentiability Homework Statement I proved in the preceding exercise the following characterization of measure zero: "A subset E of R is of measure zero if and only if it has the following property: (***) There exists a sequence J_k=]a_k,b_k[ such that every x in...

1. function is x^2 if x is rational 0 if x is irrational. I need to prove that function is only differentiable at 0. 2. f'(x)=lim(h->0)=(f(x+h)-f(x))/h 3. fruitless attempt-----> So f'(0)=lim(h->0) f(h)/h=lim(h->0)x since it's 0 when irrational and x when rational, 0 when irrational, x=0...

f(t)=(t^3, |t|^3) is a parametric representation of y=f(x)=|x|. Consider y=|x|, the left hand derivative f '-(0)=-1 and the right hand derivative f '+(0)=1, so f(x) is clearly not differentiable at 0. But f '(t)=(3t^2, 3t^2) for t>=0 f '(t)=(3t^2, -3t^2) for t<=0 f '(0)=(0,0) and f(t)...

Homework Statement Where is f(z) differentiable? Analytic? f(z) = x^{2} + i y^{2} Homework Equations Cauchy-Riemann Equations The Attempt at a Solution I calculated the partial derivatives, u_{x} = 2x v_{y} = 2y u_{y} = 0 v_{x} = 0 Then said that for the CR equations to...

So if a function f:[a,b]\to\mathbb{R} is differentiable, then then for each x\in [a,b] there exists \xi_x \in [a,x] so that f'(\xi_x) = \frac{f(x)-f(a)}{x-a} Sometimes there may be several possible choices for \xi_x. My question is, that if the mapping x\mapsto \xi_x is...

Hi there just a general question: this involves continuity and differentiability suppose: f(x) = sin 1/x if x not equal to 0 f(0) = 0 PROVE F IS NOT DIFFERENTIABLE AT 0 i understand if it is not differentiable at 0 then it may not be continuous at 0. however is there...

Homework Statement f(x) is a piecewise function defined as: |x-3| x>=1 \frac{x^2}{4}-\frac{3x}{2}+\frac{13}{4} x<1 Discuss the continuity and differentiability of this funtion at x=1 and x=3 Homework Equations The Attempt at a Solution At x=3, this function is continuous...

Theorem: Let f be continuous on [a,b]. The function g defined on [a,b] by http://tutorial.math.lamar.edu/AllBrowsers/2413/DefnofDefiniteIntegral_files/eq0051M.gif is continuous on [a,b], differentiable on (a,b), and has derivative g'(x)=f(x) for all x in (a,b) 1) Given that g is defined...

My text defines differentiability of f:M\rightarrow \mathbb{R} at a point p on a manifold M as the differentiability of f\circ \phi^{-1}:\phi(V) \rightarrow \mathbb{R} on the whole of phi(V) for any chart (U,\phi ) containing p, where V is an open neighbourhood of p contained in U. Is this...

On page 116 of Choquet-Bruhat, Analysis, Manifolds, and Physics, Lie groups are defined, and the first exercise after that asks you to prove that for a Lie group G f:G \rightarrow G; x \mapsto x^{-1} is differentiable. I know from the previous definitions that a function f on a manifold...

Let be the powe series: f(x)=\sum_{n=0}^{\infty}a(n)x^{n} then if f(x) is infinitely many times differentiable then for every n we have: n!a(n)=D^{n}f(0) (1) of course we don't know if the series above is of the Taylor type, but (1) works nice to get a(n) at least for finite n.

When you are given a function and asked for differentiability at a point in domain or the whole domain, what's the normal procedure? I think the definition is almost useless in here... Also, I'm still stuck on the previous questions so if anyone bothers to check them out...

I am proving that the function f(z) = Arg z is nowhere differentiable by using the definiton of a derivative. I let z = x + yi. Then, if the limit exists, we have f'(z) = lim (/\z -> 0) ( f(z + /\z) - f(z) ) / /\z. (Note that /\ is the triangle symbol) Also, let /\z = p + iq, where p and...

differentiability VS "derivability" In french, the quality of a function which in english is called 'differentiable', we call 'dérivable'. And we call 'différentiable' at the point (x,y) a function f such that we can write f(x+h,y+k) - f(x,y) = h*df(x,y)/dx + k*df(x,y)/dy + o(sqrt{h²+k²})...

Am mainly stuck on parts c) and d) but thought i'd put in the other questions as an aid 2. a) Define what it means to describe a function f of two real variables as differentiable at (a, b)? Define (as limits) the partial derivatives df/dx and df/dy at (a, b) and prove that if f is...

Let f(x) = x if x rational and f(x) = 0 if x is irrational Let g(x) = x^2 if x rational and g(x) = 0 if x is irrational. Both functions are continuous at 0 and discontinuous at each x != 0. How do I show that f is not differentiable at 0? How should I show that g is differentiable...

hey guys, what's up. Its my first time posting so yeah... Ok I'm attempting to do the following problem... let f(x) = x^3 for x is less than or eqaul to 4. (6x^2)-8x when x is greater than 4 i need to see if f(x) is differentiable at x = 4. I tried it through the...

Need some hints on how to go about doing this: f(x, y)=\left\{\begin{array}{cc}\frac{x^4 + y^4}{x^2 + y^2},&\mbox{ if } (x, y)\neq (0,0)\\0, & \mbox{ if } (x, y) = 0\end{array}\right. Show that f is differentiabile at (0, 0). I've tried a number of things, too ugly and not worth...

Consider f(x)=x^3-x^2+x+1 g(x)=\left\{\begin{array}{cc}{max\{f(t),0\leq t \leq x\}}\;\ 0\leq x \leq 1 \\ 3-x\;\ 1< x \leq 2\end{array}\right Discuss the continuity and differentiability of g(x) in the interval (0,2) I know how to do it As f(x) is increasing function therefore max...