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What area of math best describes continuous, but discrete changes in an object? For example.

Example. If the start vector $(1,1,1)$ changed from itself to $(1,4,2)$ in one step, then since the 2nd component jumped up by 3, that change is not continuous, where as if it went to $(1,0,2)$ it is continuous.

Now imagine if there was some final vector $(x,y,z)$. Then is there a chain of continuous changes from the start to final? In this case, yes, always. However for more exotic objects there isn't always, especially when we add the constraint that a function of the changing object must satisfy some criterion.

So for such objects, knowing whether there is a path of small discrete changes from the object to a final object is useful, and since we've limited the possible changes from one object, we have less potential changes to look at than if we were allowed to jump anywhere.

The objects I was thinking of were algorithms.

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Given that your objects are algorithms, what are the (discrete) changes? –  Quinn Culver Apr 23 '12 at 18:35
    
On second thought, applying this to algorithms directly would not work so well. –  Enjoys Math Apr 27 '12 at 16:41

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You are using what is sometimes called "The Discrete Intermediate Value Theorem." If you type $$\rm discrete\ intermediate\ value$$ into Google I think you'll find some useful links appear.

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