I know its a pretty elementary question, but I never felt like I've had any sort of reasonable explanation of why. As I understand, we can define continuity for a function f: ℝ(adsbygoogle = window.adsbygoogle || []).push({}); ^{n}→ℝ as:

For any ε>0 there exists a δ>0 such that for allxst 0< lx-al < δ then lf(x) - f(a)l < ε

Alright, so in other words, we can always define a "disc" in ℝ^{n}such that all function values belonging to the points in that disc are less that our chosen epsilon.

So let's say we find that every linear path that passes through a is continuous at the pointa. Let's also say that for each linear path, there is a maximum delta, call it δ', such that for all linear paths, for every ε we can choose δ' such that for allx0< lx-al < δ' → lf(x) - f(a)l < ε .

Doesn't this prove the continuity? I know there are counter examples that will show that if you take some other path, you may get a different limit. But it still seems to me that that the above should be true.

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# Why can't linear paths prove continuity in R^n?

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