0
votes
1answer
35 views

(Dis)continuity of function in $R^2$

$$f(x,y) = \begin{cases} a+2x^{2}-b(y-c), & x^{2}>2+x\wedge y<6\\ 3+cx-y, & else \end{cases}$$ $f(x,y)$ is continuous on $R^2$ if $a=-3, b=1, c=2$ I think it's true: insert ...
0
votes
2answers
45 views

Continuity of a multivariable function with “parts”

I'm trying to solve if $f$ is continuous: $$ f(x,y) = \begin{cases} x^3 + y^3 &\text{if }y>0 \\ x^2 &\text{if }y ≤ 0 \end{cases} $$ I have seen that $$\lim_{(x,y) \to (0,0)} ...
1
vote
1answer
23 views

What is meant by the continuity of the Hessian matrix

I have a simple and short question: "What is meant by the continuity of the Hessian matrix?" I guess it means that all the second partial derivatives of a function $f$ are continuous functions? is ...
0
votes
1answer
40 views

What am I doing wrong in this continuity check?

I want to show that the function $f$ is discontiunous. $f$ is defined as follows: $$f(x,y) := \begin{cases} (x^2+y^2)\cdot\sin(\frac{1}{\sqrt{x^2+y^2}}) & , (x, y) \neq (0,0) \\ 0 ...
1
vote
0answers
51 views

How to show $f(x)=\exp((|x|^2-1)^{-1})$ if $|x|<1$ and $f(x)=0$ if $|x|\geq 1$ is a test function?

What would be the formal argument for showing the function $f:\mathbb R^n\longrightarrow \mathbb R$, $$f(x):=\left\{\begin{array}{ccc} ...
2
votes
2answers
28 views

Epsilon delta prove for continuïty$ (1-\cos(|xy|))/y^2$

Let a function, $\mathbb{R}^2\to\mathbb{R}: \begin{Bmatrix} \frac{1-\cos(|xy|)}{y^2}&y\neq0\\ \frac{x^2}{2}&y=0 \end{Bmatrix} $ I have to prove this is continious. For y$\neq 0$, this is ...
0
votes
2answers
42 views

showing that $f(x,y)$ is continuous at $(0,0)$

Let $$f(x,y) = \begin{cases} 0, & \text{if $y \le 0$, $y \ge x^2$ } \\[2ex] 1, & \text{if $0 \lt y \lt x^2$ } \\ \end{cases}$$ Show that $f(x,y) \to 0$ as $(x,y) \to (0,0)$ along any ...
1
vote
0answers
59 views

Show that this function is continuous at $(0,0)$

In this case i'm struggling to show that the partial derivatives with respect to x are continuous. The answers always brush over how you determine it like it trivial so i think i'm missing something. ...
0
votes
1answer
19 views

Define multiple-variable function to be continuous

Define the function $f(x,y)= {{x^2 + y (x^2 + y)} \over {x^2 + y^2}}$ at $[0,0]$ so that the function would be continuous. I need help with this calculus problem. I mean, I guess it involves some ...
0
votes
1answer
32 views

Continuity proof of two-variable function.

The Assignment Determine if the following function is continuous in $(0,0)$. $$f: \mathbb{R}^2 \rightarrow \mathbb{R},\begin{pmatrix}x\\y\\\end{pmatrix} \rightarrow \begin{cases} ...
0
votes
1answer
38 views

Determine if the following function is continuous in $(0,0)$.

Assignment: Determine if the following function is continuous in $(0,0)$. $$f: \mathbb{R}^2 \rightarrow \mathbb{R},\begin{pmatrix}x\\y\\\end{pmatrix} \rightarrow \begin{cases} 1& ,x≤ 0, y ...
0
votes
0answers
59 views

Prove a two variable function to be continuos on an specific domain

Let the function $g:D \to \mathbb R^2$ be given by $$g(x,y)=2-|x+y|$$ with domain $D= \{(x,y) \in \mathbb R^2:x+y \leq2\}.$ How to prove it is continuous? I know that I need to prove this for every ...
0
votes
0answers
27 views

Show that $f: K_1(0) \rightarrow \mathbb{R}^3$ is Lipschitz.

Firstly, the Assignment: Let $V = (\mathbb{R}^3 ,\|\cdot\|_{\infty})$ where $\|\cdot\|_{\infty}$ denotes the maximum norm and consider the function: $$f: K_1(0) \rightarrow ...
15
votes
3answers
310 views

Continuity of a function in two variables

Function $f(x,y)$ is continuous in each variable separately. Prove that there exists a point where it is continuous in two variables. I do not quite understand how to act here. I know the ...
1
vote
0answers
18 views

Continuity of a piecewise defined function in two variables

I need some insight into the “approach” that I used to solve this problem. Namely, I was asked to find if the following function is continuous on all $\mathbb{R}^2$: $$ f(x, y) = \left\{ ...
1
vote
2answers
32 views

How would I finish this continuity proof?

I have a multivariable function $f$ with $$f(x, y) = \begin{cases} \frac{x^2+y^2}{y} & \text{if }y \neq 0\\ 0 & \text{if }y = 0 \end{cases}$$ and want to show that it is continuous at $(0, ...
1
vote
2answers
78 views

How to prove that $f(x,y)=3+2x+y$ is continuous?

The question is to prove that the function $f(x,y,z) = 3+2x+y$ is continuous everywhere. My approach uses the delta-epsilon method. $|(x,y)-(a,b)|\lt \delta$ then $|f(x,y)-f(a,b)|$. All I did was ...
1
vote
1answer
37 views

Checking if a piecewise defined function in two variables is continuous

How would I check if the following function is continuous? $$ f(x, y) = \left\{ \begin{array}{ll} \sqrt{1 - x^2 - y^2} & \text{, if } x^2 + y^2 \leq 1\\ 0 & \text{, otherwise} ...
0
votes
2answers
44 views

Directional derivatives of a multivariate function not defined at $(0,0)$

Let $$f(x,y)=\left\{ \begin{matrix} \frac{x^2y}{x^4+y^2} & (x,y)\neq(0,0) \\0 & (x,y)=(0,0)\end{matrix}\right.$$ It is easy to prove that the $f$ is not continuous at $(0,0)$ (doing the ...
2
votes
1answer
19 views

continous, two-dimensional function

I have a question to this two dimensional function. $f_1(x,y):=\begin{cases} \frac{2xy}{x^2+y^2},&\text{if }(x,y)\neq(0,0)\\0,&\text{else}\end{cases}$ I want to analyse if this function is ...
1
vote
2answers
77 views

Show the function is integrable and find the integral - somewhat complex question

We are given $Q = [0,1]$x$[0,1]$ We are also given the function $f(x,y) = (\frac{1}{10})^n$ where $\frac{1}{2^{n+1}} < \max(x,y) \leq \frac{1}{2^n}, (n=0,1,2,...)$ and $f(0,0)=0$. Show that $f$ ...
3
votes
1answer
41 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
42 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
33 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
23 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
1answer
33 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
96 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
57 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
20 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
71 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
64 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
44 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
61 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
58 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
132 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
44 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
49 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
96 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
31 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
122 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
80 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
43 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
56 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
62 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
64 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
184 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 ...