# Tagged Questions

Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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### Locus in $(\mathrm R ,d_{\infty})$ with $d_\infty(x,y)=\max\limits_i |x_i-y_i|$ [closed]

Find the locus of points $(x_1,x_2)$ in the plane such that their distance from $(1,2)$ is equal to $3$ at $(\mathrm R ,d_\infty(x,y)=\max\limits_i |x_i-y_i|)$ I have no idea how this is looks ...
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### Proof that the nowhere differentiable functions are dense in $C_b(\mathbb R)$.

I tried to make a proof, where I use a Weierstrass function. I was surprised at how easy it was, and thus a little doubtful as to the correctness of the proof. I've looked it over, and didn't find any ...
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### Show that if $(X, \mathcal{T}), (Y, \mathcal{J})$ are both metrizable, then $X \times Y$ with product topology is metrizable

Let $(X, \mathcal{T}), (Y, \mathcal{J})$ be both metrizable, then Claim: $X \times Y$ with product topology is metrizable I have made an attempt at this question but the notation is ...
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### How do you prove triangle inequality for this metric?

Let $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ be an increasing concave function such that $f(t) = 0$ if and only if $t = 0$. Let $(X,d)$ be a metric space. Show that $d_f = f \circ d$ defines a ...
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### Can a sequence of functions converge to a discontinuous limit under norm?

I'm a bit confused about how to take the distance between two functions where one function is discontinuous. Supposing we have the $L^1$ metric $d_1$ and $f_n(x) = x^n$ defined over $[0, 1]$. $x^n$ ...