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\chapter{Working with Discretized Functions}
\label{07-1DiscFunc}

\section{Representing Analytical Functions using Numbers}

While you're pleased with the results of your kinetics simulation, something is bothering you.  \texttt{solve\_ivp} picks the time points that it uses to solve the system of ODEs, and while you can override these points with your own using keywords, you have a more philosophical question: \underline{All you really know is these few points!}  What happens in between them?

\subsection{Sampling Converts Continuous Values to Discrete Ones}

Quantum mechanics aside, most phenomena in the universe are characterized by values that change continuously over a continuous spectrum of time.  Computers have a hard time dealing with this paradigm.  We could keep adding more and more points, but we would never reach the continuous limit, our computer would run out of storage space, and our programs would take longer and longer to run.

So, you are generally going to have to work with a discrete set of points that will \textit{represent} these continuously changing phenomena.  This process is \textbf{sampling}; each $(x,y)$ pair (or $(t,y)$, or whatever) is like a measurement of the value of the continuous function, much like a time series that you would record in an experiment.  So, if you already know the analytical form of a function (i.e. an equation that represents it) then you can simply sample it to create a \textbf{discretized function}.

\subsection{You Can't Know What You Don't Know}

Thinking about it a little more, though, you also realize that, in the case of kinetics, for example, the situation is the opposite: you are starting with a discretized function and trying to figure out more information about it.  If you could propose a functional form, then you could fit it, but what could be the functional form for the concentration of your intermediate?  

\section{Connecting the Dots: What Happens Between the Points?}

\subsection{Taylor Series}

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