# Tagged Questions

Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) ...

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### Nonexistence of a continuous injection from $\mathbb{R}^n$ to $\mathbb{R}$, for all $n \geq 2$. [duplicate]

I'm trying to do the following exercise from my lecture notes: There does not exist a continuous injection from $\mathbb{R}^n$ to $\mathbb{R}$, for all $n \geq 2$. I don't really know where to ...
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### Problem with continuity and limits in 3 dimensions

Given the function $\lim_\limits{(x,y)\to (0,0)}$ $\frac{x^2y^2}{x^4+3y^4}$ The website I was reading lecture notes from said this function is not continuous at the point in question but doesn't go ...
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### Prove the function is continious.

If the function $f(x)$ is continious at $x=0$, using definitions show that $f(rx)$ is continious at $x=0$. Here $r$ is a real number.
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### Showing Intermediate Value property and closed preimage implies continuity

Let $f : [0,1] \to \mathbb{R}$ be a function satisfying the Intermediate Value property. Assume that for any $y \in \mathbb{R},$ the preimage $f^{-1}(\{y\})$ is closed. Prove $f$ is continuous. ...
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### Equivalence of statements about a linear map

I need someone to help me solve the following exercise: Let $(X, ||\cdot||_X)$ and $(Y, ||\cdot||_Y)$ be normed vector spaces over a common field $\mathbb K$ $(\mathbb R$ or $\mathbb C)$. For a ...
### Is the given function $f$ continuous?
Problem Let $\mathbb{R}_l$ denote the reals with lower limit topology, and let $\mathbb{R}_l\times \mathbb{R}_l$ have the product topology. Then the map \$f:\mathbb{R}_l\times\mathbb{R}_l\to\mathbb{R}...