# 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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### Question about the equivalence of three versions of Closed Graph Theorem

I am studying set-valued analysis, and I saw three version of this Closed Graph Theorem. Version 1: (what I was taught in class) Let $\Gamma: X \to Y$ be a correspondence. If $\Gamma$ is closed-...
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### Prove the (path-) connectedness of the graph of a compact- and convex-valued upper hemi-continuous correspondence.

Let $\Gamma: [0, 1] \to \mathbb{R}$ be a compact- and convex-valued, upper hemi-continuous correspondence. Prove that the graph of $\Gamma$ is a connected set. Is it path-connected? This is what I ...
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### The existence of continuous approximate selection

I'm looking for a condition of the existence of continuous approximate selection for a (multi-valued) minimiser mapping. That is, let's say a continuous function $f: X \times U \to \mathbb{R}$ is ...
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### Calculate total derivative directly.

Calculate directly (not via partial differentiation) the total derivative of the function $f(x_1,x_2)=x_1^2-10x_2.$ You may wish to use the fact that $\sqrt{x^2+y^2}\geq\frac{x+y}{2}.$  For the ...
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Does this equation have a solution in the complex domain? $$\sqrt{x+3} = 3 + \sqrt{x}$$ Squaring both sides gives $\sqrt{x}=-1$, which suggests the solution to be $x=1$. But: How can the complex ...