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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 …

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 …

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 …

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 …

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 …

(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 …

In logic, this relation is called almost less than or equal, and is denoted with an asterisks on the relation symbol, like this: $f \leq^* g$. For example, the bounding number is the size of the sm …

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 …

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 …

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 …

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 …

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 $ …

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 …

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 …

Theorem. Every natural number is interesting. Proof. If not, then there are some uninteresting numbers. Let n be the least number that is not interesting. Now, that is INTERESTING! Contradiction …