# Questions tagged [multivalued-functions]

In Mathematics, set theory, a Multivalued function is defined as a left-total relation (that is, every input is associated with at least one output) in which at least one input is associated with multiple (two or more) outputs.

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### Michael's selection theorem

Michael's selection theorem states that a lower hemicontinuous multivalued map with nonempty convex closed values $\displaystyle F\colon X\rightrightarrows E$ from a paracompact space $X$ to a Banach ...
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### Is $(-1)^x$ multivalued?

$(-1)^\pi=(e^{i\pi})^\pi=e^{i\pi^2}=\cos(\pi^2)+i\sin(\pi^2)$. Wolfram Alpha lists this as the only answer. However it started with $e^{i\pi}=1$, although $e^{i\pi(2n+1)}=-1$ also for any integer n. ...
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### Help with Multivariable Delta-Epsilon Proof for $(x,y)\to (0,\pi/2)$ of $\sin(x+y)=1$ [closed]

Help with Multivariable Delta-Epsilon Proof for $$\lim_{(x,y)\to (0,\frac{\pi}{2})}\sin(x+y)=1.$$
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### What about the solutions of $z^{1/3} +1 = 0$?

I'm trying to find the zeros of the equation $$z^{1/3} +1 = 0.$$ My professor said that the solutions are the third roots of unity multiplied by $-1$. My problem is that when I calculate the cubic ...
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### Can raising a number to an irrational power have infinite solutions?

$a^{\frac{1}{2}}$ is generally considered to be the positive square root of $a$, but it also makes sense (depending on context) to consider it to be multivalued, returning all square roots of $a$ ...
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### find the coefficient of the multivariate normal distribution?

We know that the multivariate normal distribution is given by $$f(x)=\frac 1 {c} e^{-\frac 1 2(x-\mu)^T\Sigma (x-\mu)}$$ Where $c =\sqrt {\det(\Sigma)2\pi}$ How do we derive this value for $c$? EDIT:...
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### Can I get a differential of a multivariable vector field?

Say I got $\vec r=(xy,y^2,xz)$, can I have the differential $d\vec r$ ? What would be the generalization of it? I can't find in on internet, so I don't know if it's possible to do.
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### A set valued intermediate value theorem.

Does anyone know of an intermediate value theorem for set valued maps which are upper semicontinuous? Specifically, I'm looking for a theorem which says that, for example, if we are in $R^3$ and we ...
Given any function of three variables ($x,y,z$), develop a corresponding function or set of functions which, with identical inputs, can be used to generate three values ($x_1,y_1,z_1$). These values ...
The function $w:=f(z)$ defined implicitly by $\Phi(w,z)=w^2-z^2-z^3=0$ has two critical points, $z=0$ and $z=-1$. I thought both of them were branch points (and hence singularities) but I realized it'...