Another Way of Writing Piecewise Functions

Background

I changed my way of writing block equations for RSS.1 However, in my old post about the Contraction Mapping Principles, there’s an inequality on the rate of convergence of a point in a complete metric space to the unique fixed point of the Lipschitz mapping with a Lipschitz constant strictly less than one.2

\begin{aligned} d(x^*,x_n) =& d\left( \lim_{k \to \infty} x_k,x_n\right) \\\ =& \lim_{k \to \infty} d(x_k,x_n) \\\ \le& \lim_{k \to \infty} \frac{q^n}{1 - q} d(x_1,x_0) \\\ =& \frac{q^n}{1 - q} d(x_1,x_0) \end{aligned}

Starting from the second line in the above block equation, at the left of the binary relation symbols there’s a whitespace character.

Visual effects in pages under “/posts/” or the index page

Note that due to the development of Octopress, I now see three backslashes in the “MathJax Equation Source”.

It doesn’t matter much in block equations.

Problem

However, it does matter if I have to define a piecewise function. Take one defined in one of my old posts as an example.3

(Added on DEC 12TH, 2014) (Revised on SEP 3RD, 2015)

Note: As you can see from the above piecewise function, the problem is now gone.

$f(x,y) = \begin{cases} 0 &\text{if } (x,y) \in \vect{I} \land y \ge x\\\ 1 &\text{if } (x,y) \in \vect{I} \land y < x \end{cases}$

Visual effects in pages under “/posts/” or the index page

Note that due to the development of Octopress, I now see three backslashes in the “MathJax Equation Source”.

Solution

Now, I know how to tell kramdown to ignore MathJax code. This is much more convenient than the method described below.

After I observed that the two equations, which are aligned by the aligned environment, at the bottom of the post cited in footnote #3, I used the $\rm \LaTeX$ commands \left\\{ and \right. in the Markdown source file for posts to construct the left curly brace only.

f(x,y) = \left\{ \begin{aligned} 0 &\text{ if } (x,y) \in \vect{I} \land y \ge x\\\ 1 &\text{ if } (x,y) \in \vect{I} \land y < x, \end{aligned} \right.