3
votes
1answer
39 views

Continuity in $\mathbb R^n$.

we just got started with this topic today, and I am confused. Let $f:\Bbb R^2 \to \Bbb R $ with $$f(x,y) =\begin{cases} y\sin(x)/x &\text{if } x \ne 0\\ 0 &\text{else} \end{cases}$$ Now, ...
0
votes
1answer
24 views

Lipschitz continuity of $f(x,y)=4x^2+xy-\frac{1}{y-1}$ on an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace)$

Problem: Find an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace )$ which includes the points $(0, 1/2$) and $(0,3/2)$ such that the function ...
1
vote
1answer
30 views

Prove that there exists only one function f such that…

Prove that there exists only one function $$\big[f\in C\left ( \left [ 0,1 \right ],\mathbb{R} \right )s.t. f(x)=\frac{2}{5}\int_{0}^{1}(x^{2}+t^{5})f(t)dt+sin(x)\big] $$
0
votes
1answer
19 views

Continuity of a partial derivative

I have the function $$f(x,y)=\begin{cases} x^2ysin(\frac1x) & \text{if $x$ is not 0} \\ 0 & \text{if $x=0$}\end{cases}$$ And I need to find the derivative and the ...
1
vote
2answers
25 views

Multivariable calculus open set question 1

Given $[A=\left \{ (x,y)\in \mathbb{R}^{2}: x^3y^3>x^2+y^2 \right \}]$, Prove that $A$ is an open set by proving that $[A^c]$ is a closed set.
1
vote
1answer
67 views

Show a function is not continuous at a point

$$ f(x,y) = \begin{cases} \dfrac{x^2 y^4}{x^4 + 6y^8}, & \text{if }(x,y)\neq(0,0) \\ 0, & \text{if }(x,y)=(0,0) \end{cases} $$ For the definition of differentiability, I have: $$\lim_{h ...
0
votes
0answers
36 views

Totally differentiable function - definition

I know for a function of several variables, if all partial derivatives exist and they are continuous at and around a point $a$ then the function is totally differentiable at that point. I ...
0
votes
1answer
14 views

multivariable calculus question norm

Given vector space C([a,b],$ \mathbb{R} $) of continuous functions of [a,b] in $ \mathbb{R}. $ Prove that the function $ \left \| f \right \|_{1}=\int_{a}^{b}\left | f(t) \right |dt $ is a norm. Also ...
-1
votes
1answer
40 views

continuity and differentiability of function of two variables

Let $f(x,y)$ be $$f(x,y): \begin{cases} x & \text{for } y = 0\\ x-y^3\sin\left(\frac{1}{y}\right)& \text{for } y \neq 0\end{cases} $$ then check continuity and differentiability at $(0,0)$. ...
2
votes
1answer
59 views

Proving continuity of (xy)^(1/3)

Let $f(x,y)= (xy)^{\frac13}$ How would you prove that f is continous at the origin using Eplison delta argument?
3
votes
0answers
41 views

Prove that $f$ is continuous at $(0, y_0)$. where $f$ is defined on $\Bbb R^2$.

Prove that f is continuous at $(0, y_0)$ $f(x, y) = \begin{cases} (1+xy)^{1/x} &\mbox{if } x \neq 0 \\ e^y & \mbox{if } x \equiv 0. \end{cases} $ Thank you!
0
votes
1answer
47 views

Dini's Derivative of a locally Lipschitz function

Let $f:\mathbb R^n \to \mathbb R$ be a locally Lipschitz function and let $v \in \mathbb R^n$. By Rademacher's theorem we know that the gradient $\nabla f$ exists almost everywhere (i.e. it $f$ is ...
0
votes
1answer
53 views

Prove that a plane is continuous?

Okay, so the problem gives a matrix: $$ \pmatrix{ 1&2\\ 3&4} $$ and this matrix is an $\Bbb R^2 \to \Bbb R^2$ linear map. I am asked to explicitly write the component functions of $A$, and ...
5
votes
1answer
130 views

Extended matrix function

I have a continuous matrix-valued function $f:\mathbb{R}^d\mapsto {\cal M}_{k\times d}$, with $d<k$, such that $f(x)$ is full rank for all $x\in\mathbb{R}^k$. Can I extend this function to be a ...
1
vote
1answer
39 views

Why is the partial derivative $f_x' = 0 $ is not continous?

Looking again at my first CalculusII exam and I get confused about something. Let $ f(x, y) = \begin{cases} (x^2 + y^2) \sin\left(\frac{1}{x^2 + y^2}\right), & \text{if $(x, y) \ne (0, 0)$} \\ ...
2
votes
1answer
41 views

Bilinear map on the set of finite sequences

Let $X = \{ x = (x_n)_{n=1} ^ {\infty}\subset \mathbb{R} \ \ | \ \ \exists N \in \mathbb{N} : \forall n>N : x_n=0 \}$ Let the norm on $X$ be $||x|| = \sum _{n=1} ^{\infty} |x_n|$ (which is fine, ...
0
votes
0answers
52 views

calculus continuity question.please help.

Show that $\sin (x + y)$ and $\cos(x-y)$ are continuous at $(0,\pi/2)$ using $\epsilon$ and $\delta$ definition. I have tried it as follows. Let $\epsilon>0$ be any real number. To find $\delta ...
0
votes
1answer
42 views

Differentiability with non continuous partials (origin)

The function $$f(x,y)= \frac{x^{2}y^{2}}{(x^{2}+y^{4})} \quad if \quad (x,y) \neq (0,0)$$ $$f(0,0)=0$$ In order to study it's differenciability at the origin, I've studied if the partial are ...
5
votes
1answer
93 views

Prove that a mapping $f:[-1,1]^2\to\mathbb R^2$ with certain properties has the value $(0,0)$.

The mapping $f:[-1,1]^2\to\mathbb R^2$ is known to be continuous. Also the image of the upper edge of the rectangle is contained in the upper half-plane, the left edge's image is contained in the left ...
0
votes
1answer
24 views

Proving Continuity in Multiple Variables

The Exercise: $f(x,y)=xy/(x^2+y^2)$ if $x \ne 0$ $f(x,y)=0$ otherwise Where is $f$ continuous? My Attempt: At $(0,0)$, let y=kx. $lim_{(x,y)\to (0,0)}f(x,y) = lim_{x\to 0}f(x,kx)=k/(1+k^2)$ which ...
1
vote
1answer
93 views

Is $\cos\sqrt{xy}$ uniformly continuous?

I'm trying to find out if $$ f(x,y)=\cos\left(\sqrt{xy}\right) $$ is uniformly continuous on the set $\{(x, y)\in\mathbb{R}^2 : x\geq0, y\geq 0\}$. The theorems I have available to use for this are ...
7
votes
1answer
61 views

Two-variable limit, quotient of polynomials

I'm trying to evaluate the following limit, $$ \lim_{(x,y)\to(0,0)} \frac{x^3-y^2}{x^2-y} $$ which I think it doesn't exist, since for the curve $\alpha :[0,1]\to \mathbb R^2$, $\alpha(t) = (t, t^2)$ ...
2
votes
1answer
41 views

Continuosly differentation on composite functions

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ a $C^1$ function and defined $g(x) = f(\|x\|)$. Prove $g$ is $C^1$ on $\mathbb{R}^n\setminus\{0\}$. Give an example of $f$ such that $g$ is $C^1$ at the origin ...
0
votes
1answer
40 views

Question regarding Continuity of F(x,y)

Let $f(x,y) = \begin{cases} \frac{2(x^3+y^3)}{x^2+2y}&\text{ } (x,y)\not=(0,0)\\ 0 &\text{ }(x,y) =(0,0). \end{cases}$ show that first order partial derivatives of $f$ wrt x and y exist at ...
2
votes
0answers
38 views

Continuity of the orthogonal matrix-valued function

Given $d<k$. Suppose that $H:\mathbb{R}^k\rightarrow {\cal M}(\mathbb{R})_{d,k}$ is a continuous matrices-valued function such that $H(x)$ is full rank for every $x \in \mathbb{R}^k$. Now define a ...
2
votes
2answers
52 views

Continuity in multivariable calculus

I want to find out the points, where the function $f(x,y)=\dfrac{xy}{x-y}$ if $x\neq y$ and $f(x,y)=0$ otherwise, is continuous. I have shown that at all the points $(x,y)$, where $x\neq y$, $f$ is ...
0
votes
1answer
52 views

Proving non-existing limit using the path method

I have the following question: $$ f(x,y,z) = \begin{cases} {\frac{x^2 + y^2 + z^2}{|x| + y^4 + z^2};\space\space (x,y,z) \ne (0,0,0)} \\ 0\space\space ...
4
votes
3answers
63 views

Advantages to continuity at a point

A scalar field $f : \mathbb{R}^n \to \mathbb{R}$ is said to be continuous at a point $\boldsymbol{a}$ if $$ \lim_{\boldsymbol{x} \to \boldsymbol{a}} f(\boldsymbol{x}) = f(\boldsymbol{a}) $$ So in ...
0
votes
2answers
169 views

How to prove that $f(x,y)=\sqrt{x^2+y^2}$ is continuous in $\mathbb{R}^2$? [duplicate]

Please, I need the demonstration (step by step) of the continuity in $\mathbb{R}^2$ of the function $f(x,y)=\sqrt{x^2+y^2}$. I know that the function is continuous in $\mathbb{R}^2$, but I just don´t ...
1
vote
1answer
164 views

Is $f$ continuous at $(0,0)$

$$ f(x,y) = \begin{cases} \frac{xy^2}{x^2 + y^2} & \text{ if } (x,y) \neq (0,0) \\ 0 & \text{ if } (x,y) = (0,0)\end{cases} $$ (i) Is $f$ continuous at $(0,0)$? At $(x,y) \neq (0,0)$ this ...
2
votes
2answers
52 views

Making a multivariable Function continuous

This function $$f(x,y)=\frac{e^{xy}-\cos (x)+\sin(xy)}{x}$$ can be made continous for $f(0,y)$ by defining $$f(0, y) = 2y .$$ My question is: how can i get to this conclusion ("$2y$ must be it") ...
3
votes
4answers
130 views

Why Does the existence of $\frac{\partial f}{\partial x}$ not imply that $\frac{\partial f}{\partial x}$ is continuous?

For $f(x)$, the existence of $f'(x)$ implies the continuity of $f(x)$. And I am assuming that it also implies the continuity of $f'(x)$. My question is why in a function $g(x,y)$, is the existence ...
0
votes
1answer
57 views

If $\frac{\partial f}{\partial u}(u)>0$ for all $u\in S^{m-1}$, then $\exists a$ such that $\frac{\partial f}{\partial v}(a)=0$ for all $v$.

Let $S^{m-1}$ be the unit sphere in $\mathbb{R}^m$ centered at origin and $f:\mathbb{R}^m\to\mathbb{R}$ a continuous function that has all directional derivatives at any point of $\mathbb{R}^m$. Prove ...
2
votes
2answers
83 views

How to show that $f(x,y)$ is continuous.

How to show that $f(x,y)$ is continuous. $$f(x,y)=\frac{4y^3(x^2+y^2)-(x^4+y^4)2x\alpha}{(x^2+y^2)^{\alpha +1}}$$ for $\alpha <3/2$. Please show me Thanks :)
1
vote
1answer
31 views

Continuity of an integral with respect to one variable

Let $V\subseteq \mathbb{R}^n$ and $f:V\to\mathbb{R}^n$. Consider the function $$g(x_1,x_2,...,x_n) = \int_{x_2}^{x_1} {f(t,x_2,...,x_n)dt}$$ on $V$. What conditions will I need to conclude that $g$ is ...
4
votes
1answer
195 views

Limit with integral or is this function continuous?

Hello I need to show one identity and one limit. I am having problems with it. notation: $x_i$ is i-th coordinate of $x$ $B(x,r)$ ball with center $x$ and radius $r$ $S(x,r)$ sphere with center ...
2
votes
0answers
190 views

Proof on showing F(x,y) is continuous by $\epsilon - \delta$ definition

The task is as follows: Given: $$F(x,y) = \frac{xy(x^2 - y^2)}{x^2 + y^2}$$ Goal: Prove that $F(x,y)$ is continuous everywhere on the plane Here is my attempt so far: (1) By the ...
0
votes
2answers
90 views

Is this piecewise function continuous at the origin?

$f(x,y)$ is defined to be $\frac{x}{|y|}\sqrt{x^2+y^2}$ when $y \neq 0$ and $0$ when $y=0$. Is $f(x,y)$ continuous at $(x,y)=(0,0)$? I don't know why, but I can't seem to find two paths that yield ...
0
votes
2answers
60 views

Let $f:\Bbb R^2→\Bbb R:(0,0)\mapsto 0 \quad (x,y)\mapsto\frac{x^2y^2}{x^4+y^4}$

Let $$f:\Bbb R^2\to\Bbb R:(0,0)\mapsto 0 \quad (x,y)\mapsto\frac{x^2y^2}{x^4+y^4}$$ i) Is $f$ continuous at $(0,0)$? ii) Is $f$ differentiable at $(0,0)$? I can prove that $f$ is ...
0
votes
1answer
59 views

$F(x,y)$ continues at $(x_0,y_0)$?

We have $F(x,y)=f(x), f(x)$ continues at $x_0$. How to prove that: for every $y_0 \in \mathbb{R}$, $F(x,y)$ continues at $(x_0, y_0)$?
0
votes
2answers
273 views

$\epsilon$-$\delta$ proof that $f(x) = x^3 /(x^2+y^2)$, $(x,y) \ne (0,0)$, is continuous at $(0,0)$

I need to prove that $f$ continuous at $(x, y)=(0,0)$ using a $\epsilon$-$\delta$ proof $$ f(x, y) = \begin{cases} \frac{x^3}{{x^2 + y^2}},&(x,y)\neq (0,0) \\ 0,&(x,y) = (0,0) \end{cases} ...
2
votes
1answer
210 views

Continuity of a vector function through continuity of its components

Prove the following theorem: Let $f$ be a function from $\mathbb R$ to $\mathbb R^n$ and for $i=1,2,\ldots,n$ let $f_i\colon\mathbb R\to\mathbb R$ be the component functions of $f$, that is to say ...
4
votes
1answer
72 views

Is it a continuous function on $\mathbb{R}^{2}$?

Prove or disprove: let $f : \mathbb{R}^{2} \to \mathbb{R}$ be a mapping with the following properties: for each $y \in \mathbb{R}$ the function $x\mapsto f\left(x,y\right)$ is continuous on ...
1
vote
1answer
148 views

Alternative sufficient conditions for differentiability of two-variable functions?

Does anyone know of a counterexample (or proof) for the following? Suppose $f: D \to \mathbb{R}, D \subset \mathbb{R}^2$ is continuous at $(a,b)$ and its directional derivatives are linear in the ...
0
votes
1answer
240 views

Continuity of multivariable functions when “component functions” are continuous

Given topological spaces $X_1, X_2, \dotsc, X_n, Y$, consider a multivariable function $f : \prod_{i = 1}^nX_i \to Y$ such that for any $(x_1, x_2, \dotsc, x_n) \in \prod_{i = 1}^nX_i$, the functions ...
1
vote
1answer
63 views

Analyzing the continuity at a point of $f:\mathbb{R} ^{2}\rightarrow \mathbb{R}$

So i have the following function, $f:\mathbb{R} ^{2}\rightarrow \mathbb{R}$, and they ask me to analyze the continuity at the point $P = (1,0)$, when f is defined as follows: $$f(x,y) =\begin{cases} ...
1
vote
1answer
125 views

Show discontinuity in $f(x,y)$

I have $f(x,y) =\frac{2xy^2 }{x^2+y^4}$ if $(x,y) \neq (0,0)$ and $0$ if $(x,y) = (0,0)$. I figured I could just find that the limit as $\frac{2xy^2 }{x^2+y^4}$ does not equal to $0$, which should ...
2
votes
1answer
514 views

Properties of $xy^2/(x^2+y^4)$ near the origin

$f:\mathbb{R}^2\rightarrow \mathbb{R}$ Defined by $$f(x,y)= \frac{xy^2}{x^2+y^4}$$ if $x\neq 0,y\in\mathbb{R}$ and $$f(x,y)=0$$ if $x=0,y\in\mathbb{R}$ Then it is continuous but not ...