This is a link post. 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 [...]
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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