Examples of functions with linearity that are not polynomials or derivatives? Derivatives, integral and limits have this linearity feature right? Where $f(x+y) = f(x) + f(y)$ (I’m not sure if ‘linearity’ is the proper term).
Can someone give examples of more types of functions with this feature? Because the only ones I could think of were linear polynomials or something like sums.
Thank you!
 A: Good question.
Linearity is a central general property in lots of mathematics. It's the assertion that
$$
T(ax + by) = aT(x) + bT(y)
$$
whenever those operations make sense.
In the narrow question you ask the domain and codomain of $T$ are the real numbers. That's the Cauchy functional equation. The only solutions (if you add some mild continuity restriction) are the functions $T(x) = cx$. So linearity there is really just the distributive law for multiplication over addition.
In the differentiation and integration examples the domain is a space of real valued functions (so instead of $x$ and $y$ you might want to write $f$ and $g$). The constants $a$ and $b$ are real scalars. For differentiation, the codomain is again a space of functions. For definite integration (over a fixed interval) the codomain is the set of real numbers.
When you study linear algebra much of your effort will be spent understanding these transformations when $x$ and $y$ are vectors. (That's why linear algebra is called "linear algebra".
