How is addition and multiplication of step functions defined?

Question

I'm going through "Calculus" by Tom Apostol. And I'm in this section:

I think the book assumes that from the example I can extrapolate how the graph for any addition of step functions is done; nonetheless, I don't understand that example. So a problem arises now that I have to do the first exercise.

So, when I'm going to do $a)$ I know how to graph $\lfloor x \rfloor$ and $\lfloor 2x \rfloor$, I even know how to do the common refinement, but not the graph of $\lfloor 2x \rfloor$+$\lfloor x \rfloor$ itself.

So, do you think you can tell me how step functions are added and multiplied? thanks in advance.

Answer 1

Looks like you have the right idea. You divide the domain into sub-intervals, and evaluate in each sub-interval.

$f(x) + g(x) = \begin {cases} -3-6 = 7 & -3 \le x < -2.5\\-3-5 = 6 & -2.5 \le x < -2\\-2-4 = 5 & -2 \le x < -1.5\\&\vdots\end{cases}$

$f(x)g(x) = \begin {cases} (-3)(-6) = 18 & -3 \le x < -2.5\\(-3)(-5) = 15 & -2.5 \le x < -2\\(-2)(-4) = 8 & -2 \le x < -1.5\\&\vdots\end{cases}$

Answer 2

By definition functions have specific (single) values at any point. Given two functions $f$ and $g$ (it is immaterial if they are step functions or not), at any desired point $a$ in the domain get their values $f(a)$ and $g(a)$ (this may need a complicated table look up, lengthy process or could be instantaneous).

Now add these two values to get a well-defined number, and this the value of $f+g$ at $a$. This can be done for every $a$, and so we have the definition of $f+g$. In a similar way can define the product function $fg$. Definitions are conceptually simple and uniform for all kind of functions.