Differentiable Definition and 280 Threads

I've a doubt regarding Lemma 1.35 (Smooth Manifold Chart Lemma) from J. Lee "Introduction to Smooth Manifolds" The proof claims that Hausdorff property follows from v). However v) includes the case where both ##p## and ##q## are included in the same ##U_{\alpha}##, i.e. their neighborhoods are...

$$f(x,y)=\left\{\begin{array}{ccc} (x^2+y^2)\sin\left(\frac{1}{\sqrt{x^2+y^2}}\right) & , & (x,y)\neq (0,0) \\ 0 & , & (x,y)=(0,0) \end{array}\right.$$ This function is differentiable at (0,0) point but ##f_x## and ##f_y## partial derivatives not continuous at (0,0) point. I need another...

For this problem and solution, I'm confused how ##x \in (c - \delta, c + \delta)## is the same as ##0 <| x - c| <\delta##. I think it is the same as ##c - \delta < x < c + \delta## which we break into parts ##c - \delta < x \implies \delta > -(x - c)## and ##x < c + \delta \implies x - c <...

For this problem, My proof is Since ##f'## is increasing then ##x < y <z## which then ##f(x) < f(y) < f(z)## This is because, ##f''(t) \ge 0## for all t ## \rightarrow \int \frac{df'}{dt} dt \ge \int 0~dt = 0## for all t ##\rightarrow \int df' \geq 0## for all t ##f ' \geq 0## for all t...

For this problem, I'm confused by the implication from the antecedent ##0 < |x - c| < \delta## to the consequent. Should the consequent not be ##|f''(x) - f''(c)| < \frac{1}{2}## where ##\epsilon = \frac{1}{2}## (Since we are applying the definition of a limit for the first derivative curve)...

For this problem, My solution is, ##F(x)=\left\{\begin{array}{ll} e^{-\frac{1}{x}} & \text { if } x>0 \\ 0 & \text { if } x \leq 0\end{array}\right.## The we differentiate both sub-function of the piecewise function. Note I assume differentiable since we are proving a result that the function...

For this true or false problem, My solution is, With rearrangement ##\frac{f(x) - f(a)}{x - a} > f'(a)## for ##x < a## since ##f''(x) > 0## implies ##f'(x)) > 0## from integration. ##f'(x) > 0## is equivalent to ##f(x)## is strictly increase which means that ##\frac{f(x) - f(a)}{x - a} > f'(a)...

##f\left(x\right)=\begin{cases}\sqrt{\left|xy\right|}sin\left(\frac{1}{xy}\right)&xy\ne 0\\ 0&xy=0\end{cases}## I showed it partial derivatives exist at ##(0,0)##, also it is continuous as ##(0,0)## but now I have to show if it differentiable or not at ##(0,0)##. According to answers it is not...

What is the difference between an absolutely continuously differentiable function and a wave? Are all absolutely continuously differentiable equations waves?

Hey! :giggle: We consider the function\begin{align*}f:\mathbb{R}^2 &\rightarrow\mathbb{R} \\ (x,y)&\mapsto \begin{cases}\frac{x^3}{x^2+y^2} & \text{ if } (x,y)\neq(0,0) \\ 0 & \text{ if } (x,y)=(0,0) \end{cases}\end{align*} (a) Show that all directional derivatives of $f$ in $(0,0)$ exist...

Hey! :giggle: We have the function $$f(x,y)=\begin{cases}x^2\sin\left (\frac{1}{x}\right )+y^2\sin\left (\frac{1}{y}\right ) & \text{ if } xy\neq 0 \\ x^2\sin\left (\frac{1}{x}\right ) & \text{ if } x\neq 0, y=0 \\ y^2\sin\left (\frac{1}{y}\right ) & \text{ if } x= 0, y\neq 0 \\ 0 & \text{ if...

Hey! :giggle: Let's consider the function \begin{align*}f:\mathbb{R}^2&\rightarrow \mathbb{R} \\ (x,y)&\mapsto \sqrt{|x|\cdot |y|}\end{align*} Show that $f$ is partially differentiable in $(0,0)$ but not total differentiable.I have done the following: We prove that $f$ is partially...

this seems to come up frequently in undergrad math classes so it is worth asking, what is the simplest and most efficient way to show ##f(x)\in C^2(\mathbb{R})## given $$f(x)=\begin{cases} (x+1)^4 & x<-\frac{1}{2} \\ 2x^4-\frac{3}{2}x^2+\frac{5}{16} & -\frac{1}{2}\leq x \end{cases}$$ And what...

Perhaps use the definition of continuity, partial differentiability?

Problem: Show that the set of differentiable real-valued functions ##f## on the interval ##(-4,4)## such that ##f'(-1) = 3f(2)## is a subspace of ##\mathbb{R}^{(-4,4)}## This is my first bouts with rigorous mathematics and my brain is not at all wired for attacking problems like this (yet). I...

I will consider first the case of ## \left [ J \right ] = \int f(x,y,y') ##, if it is right believe is easy to generalize... $$ \Delta J $$ $$\int (f(x,y+h,y'+h'))^2 - (f(x,y,y'))^2 $$ $$\int \sim [f(x,y,y') + f_{y}(x,y,y')h + f_{y'}(x,y,y')h']^2 - [f(x,y,y')]^2$$ to first order: $$\int \sim...

Do any of you know of an article or book chapter that discusses the difference between a discontinuous potential well of length ##2L## ##V(x)=\left\{\begin{array}{cc}0, & |x-x_0 |<L\\V_0 & |x-x_0 |\geq L\end{array}\right.## and a differentiable one ##\displaystyle V(x) = V_0...

What I've tried is: I have defined a function g(x)=f(x)-x^2/2. g Differentiable in the interval [0,1] As a difference of function in the interval. so -x≤g'(x)≤1-x for all x∈[0,1] than -1≤g'(x)≤0 or 0≤g'(x)≤1 . Then use the Intermediate value theorem . The problem is I am not given that f' is...

be f Differentiable function In section [0,1] and f(0)=0, f(1)=1. so: a. f A monotonous function arises in section [0,1]. b. There is a point c∈[0,1] so that f'(c)=1. c. There is a point c∈(0,1) where f has Local max. I have to choose one correct answer.

[Moderator's note: Spin-off from another thread.] You need the structure of a topological vector field K with 0 as a limit point of K-{0}. The TVF structure allows the addition and quotient expression to make sense; you need 0 as a limit point to define the limit as h-->0 and the topology to...

I was reading about differentiable manifolds on wikipedia, and in the definition it never specifies that the differentiable manifold has a metric on it. I understand that you can set up limits of functions in topological spaces without a metric being defined, but my understanding of derivatives...

The dipole potential is given by: ##\displaystyle\psi=\int_{V'} \dfrac{\rho}{|\mathbf{r}-\mathbf{r'}|} dV' +\oint_{S'} \dfrac{\sigma}{|\mathbf{r}-\mathbf{r'}|} dS'## I need to prove that ##\psi## is differentiable at points except at boundary ##S'## (where it is discontinuous) I know if...

In mathematics, variety is a generalization of the surface idea. There are several types of varieties, according to the properties they possess. The most usual are the topological varieties and the differentiable varieties. but I still do not know what it is!

Hi, a basic question related to differential manifold definition. Leveraging on the atlas's charts ##\left\{(U_i,\varphi_i)\right\} ## we actually define on ##M## the notion of differentiable function. Now take a specific chart ##\left(U,\varphi \right)## and consider a function ##f## defined...

Homework Statement "Let ##f:ℝ^2\rightarrow ℝ## be a continuously differentiable function. Show that ##f## is not one-to-one." Homework Equations A function ##f:ℝ^n\rightarrow ℝ^m## is continuously differentiable if all the partial derivatives of all the components of ##f## exist and are...

Homework Statement Sketch the graph of the following function and use the definition of the derivative to show that the function is not differentiable at x=1. $$f(x) = \begin{cases} {-x^2+2} & \text{if } x \leq 1 \\ x & \text{if } x > 1 \end{cases}$$ Homework Equations Derivative: $$f'(x) =...

Homework Statement "Let ##f:ℝ^2\rightarrow ℝ## be defined by ##f(x,y)=\sqrt{|xy|}##. Show that ##f## is not differentiable at ##(0,0)##." Homework Equations Differentiability: If ##f:ℝ^n\rightarrow ℝ^m## is differentiable at ##a\in ℝ^n##, then there exists a unique linear transformation such...

Homework Statement Find the domain in which the complex-variable function ##f(z)=(z+1)Ln(z)## is differentiable. Note: ##Ln(z)## is the principal complex logarithmic function. Homework Equations Cuachy-Riemann Equations? The Attempt at a Solution The solution I have in mind would be to let...

<Moderator's note: Moved from a homework forum.> 1. Homework Statement I am wondering if it is the same to say : Function f is infinitely many times continuously differentiable AND Function f is infinitely many times differentiable And if it is not the same, which one defines: C^infinity...

Hello! (Wave) I want to prove that the solution $u$, $u(x,t)=\frac{1}{2 \sqrt{k \pi t}} \int_{-\infty}^{+\infty} e^{-\frac{(x-s)^2}{4kt}} \phi(s) ds, x \in \mathbb{R}, t>0 (\star)$, of the initial value problem for the heat equation, with continuous and bounded initial value $\phi$, is...

Homework Statement Homework Equations Solve using limits. Function is continuous if it's graph is continuous throughout. here the (x-1) term gets canceled in numerator and denominator. So we have a continuous graph of (x+1). The Attempt at a Solution The (x-1) term gets canceled from...

Homework Statement Prove that ##h: [0,\infty) \rightarrow \mathbb R, x \mapsto \begin{cases} x^x, \ \ x>0\\ 1, \ \ x = 0\\ \end{cases} ## is continuous but not differentiable at x = 0. The Attempt at a Solution To show continuity, the limit as x approaches 0 from the right must equal to 1...

Homework Statement Let f be ##f:[0,\infty]\rightarrow \mathbb R \\ f(x):= \begin{cases} e^{-x}sin(x), \ if \ \ x\in[2k\pi,(2k+1)\pi] for \ a \ k \in \mathbb N_0 \\ 0 \ \ otherwise\\ \end{cases}## Exercise: Determine all inner points of the domain where f is also differentiable and determine...

Hey! :o I want to show that series $$f(x)=\sum_{k=1}^{\infty}2^k\sin (3^{-k}x)$$ is continuously differentiable. We have that $|2^k\sin (3^{-k}x)|\leq 2^k\cdot 3^{-k}=\left (\frac{2}{3}\right )^k$, or not? The sum $\sum_{k=1}^{\infty}\left (\frac{2}{3}\right )^k$ converges as a geometric...

Homework Statement Prove that the function ## f(z)= 1/\sqrt{2}(\sqrt{\sqrt{x^{2}+y^{2}}+x}+i*sgn(y)\sqrt{\sqrt{x^{2}+y^{2}}-x})## is holomorphic on the domain ## \Omega = \left \{ z: z \neq 0, \left | \arg{z} \right | <\pi\right \} ## and further that in this domain ##f(z)^{2} = z. ##...

Hi, are there any models known in QM where the wavefunctions do not have to be infinitely differentiable, and thus can exist in other spaces than the Hilbert space? I assume Banach spaces allow elements that are not infinitely differentiable as subsets. Can therefore certain phenomena in QM be...

Homework Statement Homework Equations All polynomial functions are continuous so the function is continuous everywhere. For differnetiable we differentiate the polynomial. But how to do this? The Attempt at a Solution I only removed option A as polynomial functions are always continuous. How...

Determine, with proof, all the real-valued differentiable functions $f$, defined for real $x > 0$, which satisfy $f(x) + f(y) = f(xy)$ for all $x, y > 0$.

Homework Statement [/B] 2. The attempt at a solution I'm not really sure where to start. We just want to show that ##\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = 0##. I see that ##\lim_{x \to c} (x - c)^2 = 0##. I feel that this may be a simple trick of inequalities, but I am having a complete...

I want to figure out whether the functions are differentiable at c. I think I should use some of the trig identities, but I'm not sure which ones. Any tips?

Homework Statement Find the values of a and b that make f a differentiable function. Note: F(x) is a piecewise function f(x): Ax^2 - Bx, X ≤ 1 Alnx + B, X > 1 Homework EquationsThe Attempt at a Solution Made the two equations equal each other. Ax^2 - Bx = Alnx + B Inserting x=1 gives, A - B =...

Homework Statement Let ##f(t)=\int_{t}^{t^2} \frac{1}{s+\sin{s}}ds,t>1.##Express ##f## as a composition of two differentiable functions ##g:ℝ→ℝ^2## and ##h:ℝ^2→ℝ##. In addition, find the derivative of ##f## (using the composition). Homework EquationsThe Attempt at a Solution Honestly, I have...

Homework Statement Let ##f##: ##G\subset\mathbb{R}^2\rightarrow\mathbb{R}## be differentiable at ##(x_0,y_0)\in{G}## and ## \lim_{(x, y) \to (x_0, y_0)} \frac{f(x,y) -a -b(x-x_0) -c(y-y_0)}{\sqrt{(x-x_0)^{2} + (y-y_0)^{2}}} = 0.## The task is to prove that then ##a=f(x_0,y_0),b=f_x(x_0,y_0)##...

Hello all, I am not sure how to approach this question: Let f(x) be a continuous and differentiable function of order 2. Let f''(x) >0 for all values of x. The tangent line to the function at x=1 is y=-x+1. Show that f(1.1)>-0.1. Thanks!

"If a function can be differentiated, it is a continuous function" By contraposition: "If a function is not continuous, it cannot be differentiated" Here comes the question: Is the following statement true? "If a function is not right(left) continuous in a certain point a, then the function...

Hello! (Wave) Suppose that we want to check if $f(x,y)=\frac{x}{y}+\frac{y}{x}$ is differentiable at each point of its domain and if it is $C^1$. The domain is $D=\{ (x,y) \in \mathbb{R}^2: x \neq 0 \text{ and } y \neq 0\}$.The partial derivatives ...

Suppose a certain function in continuous at c and (c. f(c)) exists, then which of the two could be false: \displaystyle \lim_{x \rightarrow c^-} {f(x)} = \lim_{x \rightarrow c^+} {f(x)}, and \displaystyle f'(c)? I feel like both could be false, because if the formal derivative at a point...

Since lnx is defined for positive x only shouldn't the derivative of lnx be 1/x, where x is positive. My books does not specify that x must be positive, so is lnx differentiable for all x?

Hello mates, is the function ## f(x) = \frac{(2^x - 1)}{x} ## differentiable at x = 0? For it to be differentiable it has to be continuous? From the graph f(0) is undefined although limit exists. I have read that at points like a corner, gap and vertical tangents it is not differentiable. So...

Are all real life signals infinitely continuous and differentiable? I'm thinking yes because a finite discontinuity in one of the derivatives would imply infinite to take place in the next higher-order derivative. And infinite means infinite energy.