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Let$ f : [a, b] →\mathbb R$. By the graph of $f$ we mean the set $F = \{(x, f(x)) : a ≤ x ≤ b\} ⊆ \mathbb R^2$

Prove that $f$ is continuous if and only if the graph of f is a pathwise connected subset of the plane.

I understand that a metric space is pathwise connected if for every $x,y$ in the metric space, there exists a continuous map $f$ from the unit interval $[0,1]$ to $X$ such that $f(0) = x$ and $f(1) = y$.

Going forwards is relatively simple, but I'm not sure how to prove it backwards.

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jane
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1 Answers1

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Backwards. Let $x_n\to x\in [a,b]$ and $\|(x_n,f(x_n))-(x,f(x))\|<1/n$ (you can construct a sequence like this because $F$ is path-connected). Thus $f(x_n)\to f(x)$ and so $f$ is continuous

Haha
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