# Tagged Questions

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### $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$be continuous and define$g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ by $g(x)=|f(x)|$ prove g is continuous

This was assigned as a practice problem for my multivariable calculus class, and its really not making sense to me, can someone help me out Let $f: \mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ be a ...
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### continuity single variable function and multivariable funtion and its parcial derivatives

Is f(x)=1/x discontinuous at point x=0 or not since its domain is x>0 and x<0? And what about f(x,y)=$\frac{xy^2}{x^2+y^2}$ continuity? And Df(x,y) exist or parcial derivatives are ...
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### Show that the function $f(\textbf{x}) =|\textbf{x}|$ is continuous on $\mathbb{R}^n$

I can see this intuitively, but looking for a solid answer with reasoning. all ideas will be appreciated,
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### Continuity of piecewise function

$$f(x,y) = \begin{cases} \dfrac{\sin(xy)}{xy} & \text{if x y \ne 0} \\ 1 & \text{if xy=0} \end{cases}$$ all ideas are appreciated i think this is non-continuous, i did by converting to ...
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### Proof of the rank theorem in Rudin's PMA book

I am studying Rudin's proof of the rank theorem (theorem 9.32 in Principles of Mathematical Analysis.) We have an invertible function $H(x)$ defined on an open set. He claims we can "shrink" the open ...
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### (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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $f$ ...
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### 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, ...
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### 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 ...
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### Prove that there exists only one function f such that…

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