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

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

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

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

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

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