From: Zack M. Davis Date: Thu, 16 Jul 2026 22:30:19 +0000 (-0700) Subject: matrix display fix X-Git-Url: https://zackmdavis.net/blog/source?a=commitdiff_plain;h=a6e4db32e83ea7ccf14d7ac56bec99d8ac1d9458;p=An_Algorithmic_Lucidity.git matrix display fix --- diff --git a/content/2025/discontinuous-linear-functions.md b/content/2025/discontinuous-linear-functions.md index ce06e34..e18ccae 100644 --- a/content/2025/discontinuous-linear-functions.md +++ b/content/2025/discontinuous-linear-functions.md @@ -10,7 +10,7 @@ We know what continuity is. A function _f_ is continuous iff for all ε there ex 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 crazy. I've seen many linear functions in my time, and they were definitely all continuous. _f_(_x_): ℝ → ℝ := _ax_ is continuous for any _a_ ∈ ℝ. _T_(**x⃗**): ℝ² → ℝ² := $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \boldsymbol{\vec{v}}$ is continuous no matter what the entries in the matrix are. Stop being crazy!! +Sometimes people talk about discontinuous linear functions. You might think: that's crazy. I've seen many linear functions in my time, and they were definitely all continuous. _f_(_x_): ℝ → ℝ := _ax_ is continuous for any _a_ ∈ ℝ. _T_(**x⃗**): ℝ² → ℝ² := $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \boldsymbol{\vec{v}}$ is continuous no matter what the entries in the matrix are. Stop being crazy!! Actually, it's not crazy. It's just that all the discontinuous linear functions live in infinite-dimensional spaces.