Differentiability Definition and 197 Threads

I am currently working through Nakahara's book, "Geometry, Topology and Physics", and have reached the stage at looking at calculus on manifolds. In the book he states that "The differentiability of a function f:M\rightarrow N is independent of the coordinate chart that we use". He shows this is...

Hi If the function ##f(x,y)## is independently continuous in ##x## and ##y##, i.e. f(x+d_x,y) = f(x,y) + \Delta_xd_x + O(d_x^2) and f(x,y+d_y) = f(x,y) + \Delta_yd_y + O(d_y^2) for some finite ##\Delta_x##, ##\Delta_y##, and small ##\delta_x##, ##\delta_x##, does it mean that it is continuous...

In complex analysis differentiability for a function ##f## at a point ##z_0## in the interior of the domain of ##f## is defined as the existence of the limit $$ \lim_{h\rightarrow{}0}\frac{f(z_0+h)-f(z_0)}{h}.$$ But why are the possible ##z_0##'s in the closure of the domain of the original...

(a) State precisely the definition of: a function f is differentiable at a ∈ R. (b) Prove that, if f is differentiable at a, then f is continuous at a. You may assume that f '(a) = lim {f(x) - f(a)}/(x - a) x→a (c) Assume that a function f is differentiable at each x∈ R and...

Before asking a question I would first like to mention the definitions of limit of function and differentiality at x=p 1) Limit of function (f) at x=p Let E be domain of f and p be a limit point of E. Let Y be the range of f. If there exists q∈E such that for all ε>0 there exists δ>0...

Homework Statement Discuss the continuity, derivability and differentiability of the function f(x,y) = \frac{x^3}{x^2+y^2} if (x,y)≠(0,0) and 0 otherwise Homework Equations if f is differentiable then ∇f.v=\frac{∂f}{∂v} if f has both continuous partial derivative in a neighbourhood of x_0...

Hey guys, More questions for you guys this time, these seem easy but always have a few nuances I seem to miss. With that said, I'd greatly appreciate your guys' help. Question: For 1a, I sketched two straight lines where x=/ 1 and y=1/2 for one line and y= -1/2 for the other. Thus, x=1 is...

Theorem: ctsly differentiable at a if the function is cts and its partial derivatives exist and are cts in a neighborhood of a. [1] - so to be differentiable we can check whether this conditions holds, and if it does ctsly diff => diff. - the definition of a scalar function being...

I guess my first questions is whether saying that a function is differentiable is the same as saying that its derivative is continuous. i.e. if \lim_{x\rightarrow{}a}f'(x)=f'(a) then the function is differentiable at ##a##. Or is it just a matter of the value ##f'(a)## existing? Now my...

Hi, I'm having trouble évaluation the differentiability at (0,0) of the function f(x,y)=\frac{x^3}{x^2+y^2} for (x,y) not nul, and f(x,y)=0 if (x,y)=0 I know f is differentiable if (x,y) isn't nul since the partial derivative are continuous, but I don't know how to evaluate it at (0,0)...

I have found a question Prove that f(z)=Re(z) is not differentiable at any point. According to me f(z)=Re(z)=Re(x+iy)=x which is differentiable everywhere. Then where is the mistake?

Homework Statement Discuss the continuity and differentiability of f(x) = \begin{cases} x^2 & \text{if } x\in \mathbb{Q} \\ x^4 & \text{if } x\in \mathbb{R}\setminus \mathbb{Q} \end{cases} Homework Equations The Attempt at a Solution From the graph of ##f##, I can see...

Homework Statement I have the function f, defined as follows: f=0 if xy=0 f= ##xysin(\frac{1}{xy})## if ##xy \neq 0## Study the differentiability of this function. The Attempt at a Solution there are no problems in differentiating the function where ##xy\neq0##. the partials in (0,0)...

Homework Statement . Let ##f:\mathbb ℝ:→ℝ^2## be a function defined as: ##f(x,y)=\frac{x^2y-2xy+y} {(x-1)^2+y^2} \forall (x,y)≠(1,0)## and ##f(1,0)=0##. Prove that for any curve ##α:(-ε,ε)→ℝ^2## of class ##C^1## (where ##ε>0##) such that ##α (0)=(1,0)## and ##α(t)≠(1,0)## for every ##t≠0##, the...

Homework Statement http://i.minus.com/jbzvT5rTWybpEZ.png Homework Equations If a function is differentiable, the function is continuous. The contrapositive is also true. If a function is not continuous, then it is not differentiable. A function is differentiable when the limit definition...

f:\mathbb{C}\rightarrow\mathbb{C} \\ f(z)=\left\{\begin{array} \frac{(\bar{z})^2}/ {z} \quad z\neq0 \\ 0 \quad z=0 \end{array} \right. Show that f is differentiable at z=0, but the Cauchy Riemann Equations hold at z=0. Well i have tried to start the first part but i am stuck, could you...

Homework Statement f(x) = sin ∏x/(x - 1) + a for x ≤ 1 f(x) = 2∏ for x = 1 f(x) = 1 + cos ∏x/∏(1 - x)2 for x>1 is continuous at x = 1. Find a and b Homework Equations For a lim x→0 sinx/x = 1. The Attempt at a Solution I tried...

Assume that a point x is an interior point of domain of some function f:[a,b]\to\mathbb{R}, and assume that the limit \lim_{(\delta_1,\delta_2)\to (0,0)} \frac{f(x+\delta_2)-f(x+\delta_1)}{\delta_2-\delta_1} exists. What does this imply? Well I know it implies that f'(x) exists, but...

Let f: R2-->R be defined by f(x,y) = xy2/(x2+y2 if (x,y) ≠ 0, f(0,0) = 0 a) is f continuous on R2? b) is f differentiable on R2? c) Show that all the dirctional derivatives of f at (0.0 exist and compute them Attempt: a) I had an idea to show that multivariate functions are...

Homework Statement Determine if f(x,y) = ((x-y)4 +x3 +xy2)/(x2+y2) [f(x,y = 0 @ (0,0)] is differentiable at the origin. Homework Equations x = (0,0) The Attempt at a Solution A function is differentiable at x if f(x+Δx) - f(x) = AΔx + |Δx|R(x) Where A are constant...

Homework Statement So I'm doing problems where I have to verify Rolle's hypotheses. I am only having trouble with the differentiability part. My professor wants me to prove this. So for example, f(x)=√(x)-(1/3)x [0,9] Homework Equations none The Attempt at a Solution 1.) I know the...

a)Does convergence imply being properly defined? So would it not be properly defined if it was divergent? b)I am having trouble why the last part (in the attachment) says, "Then, by (1), f(x_0) - f(x) \geq dfrac{2^k} for all [itex]x < x_0." But does (1) tell us that it's "equal" instead of...

As a preface, this question is taken from Vector Calculus 4th Edition by Susan Jane Colley, section 2.3 exercises. Homework Statement "Explain why each of the functions given in Exercises 34-36 is differentiable at every point in its domain." 34. xy - 7x^8y^2 cosx 35. \frac{x + y +...

Hello, How do I solve this kind of problems ? For which values of x the next function is "differentiable" ? I know it has something to do with the existent of the one sided limits, but which limits should I be calculating exactly ? Thanks !

Hi, Homework Statement (I) The following function is defined for α,β>0: f(x) = { xβsin(1/xα), x≠0; { 0, x=0 I was asked for the values of α,β for which f(x) would be continuous at 0, differentiable at 0, continuously differentiable at 0, and twice differentiable at 0. (II) I was asked to...

I'm working on a problem where I need to show that the series of functions, f(x) = Ʃ (xn)/n2, where n≥1, converges to some f(x), and that f(x) is continuous, differentiable, and integrable on [-1,1]. I know how to show that f(x) is continuous, since each fn(x) is continuous, and I fn(x)...

f: R2 to R1 given by f(x,y) = x(|y|^(1/2)) show differentiable at (0,0) so I'm using the definition lim |h| ->0 (f((0,0) + 9(h1,h2)) - f(0,0) - Df(0,0) (h1,h2)) / |h| so first for the jacobian for f, when I'm doing the partial with respect to y, do I have to break this into the case y>0...

Homework Statement http://i.imgur.com/69BmR.jpg Homework Equations The Attempt at a Solution a, c are right because f(c) is continuous. b, d are right because f'(c) is differentiable over the interval I am not sure about e. Can anyone explain to me?

Could someone explain this to me in terms of limits and derivatives instead of plain english? For example, how would you solve a question that says find whether the function f is differentiable at x=n and a question that asks find whether the function f is continuous at = n...

For a function of a single variable, I can check if the function is differentiable by simply taking the limit definition of a derivative and if the limit exists, then the function is differentiable at that point. Differentiability also implies continuity at this level.Now, for a function of...

Hello, I'm having problems figuring out theoretical problem on "differentiability of a function". [I hope that I spelled it right...] Suppose that: 1. Functions f(x,y) and g(x,y) are well defined in some little domain around (0,0). (*1) 2. g(x,y) is continuous at (0,0). (*2) 3. f(x,y)...

Homework Statement Refer to attached file. The attempt at a solution (a) g'(0) = \lim_{x\rightarrow 0} {\frac{g(x)-g(0)}{x-0}} g'(0) = \lim_{x\rightarrow 0} {\frac{x^\alpha cos(1/x^2)-0}{x}} g'(0) = \lim_{x\rightarrow 0} {x^{(\alpha-1)} cos(1/x^2)-0} g'(0) = 0 So...

Hi, I have a small question about this. Using the chain rule, I know that a composition of differentiable functions is differentiable. But is it also true that if a composition of functions is differentiable, then all the functions in the composition must be differentiable? For example, if...

Homework Statement Show that the function f defined by f(z) = 3\,{x}^{2}y+{y}^{3}-6\,{y}^{2}+i \left( 2\,{y}^{3}+6\,{y}^{2}+9\,x \right) is nowhere differentiable.The Attempt at a Solution Computing the C.R equations for this, I am left with {y}^{2}+2\,y={\it xy} and x^2+(y-2)^2 = 1...

I have a question regarding functional differentiablility. I understand that Frechet differntiability of a functional T with respect to a norm \rho_1 implies Hadamard differentiability of the functional T with respect to the same norm. However, it is no surprise that there would be cases...

Homework Statement let the function f:ℝ→ℝ be differentiable at x=0. Prove that lim x→0 [f(x2)-f(0)] ______________ =0 x Homework Equations The Attempt at a Solution I am kind of lost on this one, I have tried manipulating the definition of a...

Homework Statement State the Cauchy-Riemann equations and use them to show that the function defined by f(z) = |z|^2 is differentiable only at z = 0. Find f′(0). Where is f analytic? The Attempt at a Solution f(z) = |z|^2 = (x^2 + y^2) \frac{du}{dx} = 2x, \frac{dv}{dy} = 0 \frac{du}{dy} =...

The paragraph says, " Even if the function f is an everywhere differentiable function, it is still possible for f ' to be discountinuous. However, the graph of f ' can never exhibit a discountinuity of ..." picture is in paint document... What type of discountinuity is that? a hole...

Since I am new to PF (hi!), before I go any further, I would like to a) briefly note that this is an independent study question, and that its scope goes beyond that of a textbook question - i.e., I believe that this thread belongs here - and b) also note that I am new to analysis and early...

I'm trying to prove that if a function is continuous on [a,b] and smooth on (a,b) then there's a point x in (a,b) where f'(x) exists. The definition of smoothness is: f is smooth at x iff \lim_{h \rightarrow 0} \frac{ f(x+h) + f(x-h) - 2f(x) }{ h } = 0 . I'm starting with the simpler case...

Are |x| and [x] differentiable anywhere? If so, what're their derivatives?

as you know i have been asked a question which no no way i couldn't tackle it. and its is about differentiabilty. at long last i found a solution. i want to share with you. could you check out please. thanks for now.this is the question. and this is my solution.(i assume that when x goes...

i tried to solve this problem. i can do it a little. but i can't progress. as far as I'm concerned, it requires outstanding performance. thanks for now... PROBLEM MY SOLUTION...

i need your helps. thanks for everything.

So the way I understand complex differentiability and its requirement that the partial derivatives satisfy the Cauchy-Riemann Equations is that we would really like ℂ to have the same nice property as ℝ, that is to say we would really like the derivative to be a linear operator which is itself...

I've worked all of these out. I'm mostly confident I did them correctly, but I'm prone to overlook subtleties or counterexamples sometimes. http://i111.photobucket.com/albums/n149/camarolt4z28/1ab-1.png http://i111.photobucket.com/albums/n149/camarolt4z28/1gf2-1.png

Homework Statement If f(x) = 3 for x < 0 and f(x) = 2x for x ≥ 0, is f(x) differentiable at x = 0? State and justify why/why not. Homework Equations The Attempt at a Solution Obviously, since f(x) is not continuous and the limit doesn't exist as x\rightarrow0, the function...

1. Suppose f(x)=0 if x is irrational, and f(x)=x if x is rational. Is f differentiable at x=0? 2. the derivative= lim[h->0] [f(a+h)-f(a)]/h 3. I don't really know how to start, but I do know that between any two real numbers, there exists a rational and irrational number. So I'm...

Homework Statement B(x)= xsin(1/x) when x is not equal to 0 = 0 when x is equal to 0 Determine if the function is differentiable at 0 Homework Equations The Attempt at a Solution I get B'(x)= sin(1/x)+cos(1/x)*(-1/x) but really do not know what should be done next...

Homework Statement For which parameter \alpha\in\mathbb{R} the function: f(x, y)= \begin{cases}|x|^\alpha \sin(y),&\mbox{ if } x\ne 0;\\ 0, & \mbox{ if } x=0\end{cases} is differentiable at the point (0, 0)? The Attempt at a Solution For α<0, the function is not continue at (0, 0)...