[Linkpost] “Discontinuous Linear Functions?!” by Zack_M_Davis

We know what linear functions are. A function f is linear iff it satisfies additivity <span>_f(x + y) = f(x) + f(y)_</span> and homogeneity <span>_f(ax) = af(x)_</span>.

We know what continuity is. A function f is continuous iff for all ε there exists a δ such that if <span>_|x - x_0|_</span> < δ, then <span>_|f(x) - f(x_0)|_</span> < ε.

An equivalent way to think about continuity is the sequence criterion: f is continuous iff a sequence <span>_(x_k)_</span> converging to <span>_x_</span> implies that <span>_(f(x_k))_</span> converges to <span>_f(x)_</span>. That is to say, if for all ε there exists an N such that if k ≥ N, then <span>_|x_k - x|_</span> < ε, then for all ε, there also exists an M such that if k ≥ M, then <span>_|f(x_k) - f(x)|_</span> < ε.

Sometimes people talk about discontinuous linear functions. You might think: that's [...]

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First published:

June 6th, 2025

Source:

https://www.lesswrong.com/posts/GodqHKvQhpLsAwsNL/discontinuous-linear-functions

Linkpost URL:
http://zackmdavis.net/blog/2025/06/discontinuous-linear-functions/

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Narrated by TYPE III AUDIO.