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The answer is yes. Every function on the reals is the sum of two injective functions, and this can be done in a highly effective manner, constructing the two functions $g,h$ from $f$ without any need …

There is some fascinating work in the subject of cardinal characteristics of the continuum in set theory that directly relates to the concept of growth rates of functions. I believe that it is the ide …

The other answers are excellent, but let me add a few points. First, with a historical perspective, all the early fundamental theorems of calculus were first proved via methods using infinitesimals, r …

The concept of definable real number, although seemingly easy to reason with at first, is actually laden with subtle metamathematical dangers to which both your question and the Wikipedia article to w …

There are a truly enormous number of solutions, if one only wants the solution to work on an interval. Indeed, one can find solutions to $f(f(x)) = g(x)$ for any function $g$ defined on an interval. S …

Perhaps one of the earliest examples would be with the Pythagoreans, who held that any two magnitudes were commensurable, measured as integer multiples of a smaller common unit, a belief that was conn …

One of the most well-known examples is Frege's axiom of general comprehension, which asserts that for any definite property $P$, one may form the set $$\{x\mid\ P(x)\ \},$$ consisting of all objects $ …

(From the post on my blog:) To my way of thinking, this is a serious question, and I am not really satisfied by the other answers and comments, which seem to answer a different question than the one …

All the various computer algebra system approaches to calculus, which are increasingly prevalent and powerful, although imperfect, seem to me to be prime instances of the kind of foundations you menti …

One cannot prove the Archimedean property for the reals by appealing only to first-order algebraic truths of the ordered real field and the subring of the integers sitting inside it. The reason is tha …

The distance function to a closed set is continuous, even Lipschitz continuous, and is zero exactly on that closed set. A modified version of this function can be made continuously differentiable, by …

Although you asked about continuous functions, here is an example of a discontinuous function $f:[0,1]\to\mathbb{R}$ which is not monotone on any measurable set with positive measure. Let $V\subset …

Let me assume that you mean the order on functions $f$ and $g$ by which $f\leq g$ if and only if $\exists C\exists x_0\forall x\geq x_0$ $f(x)\leq C\cdot g(x)$. In other words, $f(x)$ is eventually le …

The function is easily seen to be Borel, since the graph of the function can be defined using only natural number quantifiers. In particular, a number is in the support if and only if there is a last …

I don't know the answer to your question, but perhaps we might gain insight from the following Interview with Jason Simmons, a professional rock/paper/scissors player, which appeared a few years ago o …