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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!

user170231
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proof-of-correctness
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    "Additive" seems to be the word you're looking for – user170231 Oct 07 '22 at 17:27
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    @Ibrahim What kind of functions are you looking for? Are you specifically considering functions over the real numbers, or can these also be functions of vector spaces (as in $f:\Bbb R^m \to \Bbb R^n$)? – Ben Grossmann Oct 07 '22 at 17:44
  • @BenGrossmann Well I was looking for functions over reals; but any example of those would be cool too! – proof-of-correctness Oct 07 '22 at 17:52
  • Functions from the reals to the reals that satisfy this property either have the form $f(x) = ax$ for some real constant $a$ or are very pathological -- if not a constant times $x,$ then the function is discontinuous at every point, and even worse, it's graph is a dense subset of ${\mathbb R}^2$ and even much worse than that (because even a Baire $2$ function can have a dense graph, and nonlinear additive functions are not even Lebesgue measurable, much less Borel measurable). – Dave L. Renfro Oct 07 '22 at 18:03

2 Answers2

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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".

Ethan Bolker
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The trace is a linear map from $n\times n$ matrices to $\mathbb{R}$:

$$\text{tr}(c_1A+c_1B)=c_1\text{tr}(A)+c_2\text{tr}(B).$$

for scalars $c_1,c_2$.

You can read more about linear maps here.

Golden_Ratio
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