1
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1answer
26 views

A “repeated roots allowed” version of the continuity of roots

Let $R_n$ denote the set of all monic real polynomials of degree $n$ all of whose roots are real. Then $R_n$ is a closed subset of the $n+1$-dimensional space ${\mathbb R}_n[X]$. For $P\in R_n$, ...
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
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2answers
74 views

Basic B-Spline basis function question

I am studying the basic recursion formula for generating B-Spline basis functions N(i,j) of a given degree from the basis for the lower degree, and puzzling at the magic. In particular what I am ...
0
votes
1answer
27 views

Determine all the $x_0$ such that $\phi : \mathbb C[X] \to \mathbb C, P \mapsto P(x_0)$ is continuous

In $\mathbb C[X]$, we consider the norm $\left\lVert P \right\rVert = \sup \left|a_i\right|$ for $P(X) = \sum_{i=1}^na_ix^i$. For all $x_0$ we consider the linear form $\phi : \mathbb C[X] \to \mathbb ...
0
votes
2answers
127 views

A quadratic form is continuous on $\mathbb R^n$

Prove that the following quadratic function is continuous in $\mathbb{R}^{n}$, ...
119
votes
3answers
3k views

A Topology such that the continuous functions are exactly the polynomials

I was wondering which fields $K$ can be equipped with a topology such that a function $f:K \to K$ is continuous if and only if it is a polynomial function $f(x)=a_nx^n+\cdots+a_0$. Obviously, the ...
4
votes
2answers
91 views

Application of Bolzano's theorem to polynomials

I am trying to solve the following: Let $f$ be a polynomial of degree $n$, say $f(x)=\sum\limits_{k=0}^nc_kx^k$, such that the first and last coefficients $c_0$ and $c_n$ have opposite signs. ...
5
votes
1answer
181 views

If $p$ is a non-zero real polynomial then the map $x\mapsto \frac{1}{p(x)}$ is uniformly continuous over $\mathbb{R}$

Let $p(x)$ be a non-constant polynomial with real coefficients such that $p(x) \neq 0$ for all $x \in \mathbb{R}$. Define $f(x)=\frac{1}{p(x)}$ for all $x \in \mathbb{R}$. Prove that, for ...
0
votes
0answers
81 views

Isolation of zeros in the case of univariate analytic functions expressed as a bivariate function.

We know that the zeros of an analytic non-constant function are always isolated. A proof is here. Let $L(v)$ be an analytic function in $v$, where $v\in\mathbb{R}$. Let us write $L(v) \equiv L(v,p)$ ...
1
vote
0answers
56 views

Lower-bounding the distance between zeros of a continuous function

Consider a continuous function of the form: $L(v) = \sum_{i = 0}^{m}[vA_{i} - B_{i}]p^{i}$ where $p$ is the root of the polynomial equation: $vf(p) - g(p) = 0$ with $f(p)$ and $g(p)$ being two ...
2
votes
1answer
75 views

Are the following linear maps continuous ?

I am supposed to find the whether the following maps are continuous or not , if continuous then to find the $||T||$ $P$ is a vector space of polynomials . Define norm on the polynomials $p\in P$ as ...