Suppose and are functions , and consider the piecewise function

For no particularly good reason, today I went in search of another way to write this expression. I wasn’t sure exactly what sort of expression I was shooting for, but I hoped I’d know it when I saw it.

I’m want to try to avoid explicitly writing a piecewise function, and somehow use to disguise the piecewise nature of the function. This can be further obfuscated by writing it as , but I’ll leave it as . Actually, a function I think might be more useful is , a favorite from calculus classes. Recall that this is 1 for positive, and -1 for negative. Conveniently, the domains of the pieces here are exactly the domains of the pieces I want for .

My first step actually makes looks worse:

However, let me write and . Then notice that the pieces in the above line are exactly or . That is,

I can wrap this up using my from above to get that sign distinction, arriving at

That’s a reasonable answer I suppose. I’ve written my piecewise function all on one line, I guess, as

With this, I could then shift things horizontally and assume that my pieces where defined on and , instead of just splitting at 0. I could also, sort of inductively, get a piecewise function with more than 2 pieces. To do this, just split it into a leftmost non-piecewise function, and then a right-hand piecewise function, and use the ideas above.

I don’t think that this is the answer that makes me the happiest though. But it’s the only idea I’ve got currently. The reason I don’t like this is that and need to be defined on all of , even though in the original piecewise function they need only be defined on half the line. Also, I’m not sure I’m a huge fan of the handling of more than 2 pieces with the above method.

So, what suggestions have you got for improvement?

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March 17, 2009 at 1:05 pm |

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