Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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On an injective ring homomorphism from the ring of continuous functions to the ring of differentiable functions

Let $\phi : C \to D$ be an injective ring homomorphism such that $\phi(1)=1$, where $1$ denotes the constant function $1$ and $C,D$ are the rings of continuous and differentiable functions on ...
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Are any tools or techniques available to solve the “placement of safety points” problem?

Definition 0. Given a metric space $X$ and subsets $H$ and $S$ thereof, define: $$d(H,S) = \sup_{h \in H} \inf_{s \in S}d(h,s)$$ Here's some slightly dodgy intuition. Imagine $S$ is a set of ...
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Solving or knowing something about a non-linear PDE which is “almost” linear?

Let $a>0$ be fixed. I have the following PDE: $u=u(t,x)$, $t\in [0,1]$, $x\in \mathbb{R}$, $$-\partial_t u = |\partial_x u| + \frac{1}{2}\partial_x^2 u, \quad ...
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algebraic sum of a graph of continuous function and itself - measure > 0 imply nonempty interior?

Let $f\colon[0,1]\to\mathbb{R}$ be a continuous function. Let $G\subset\mathbb{R}^2$ be a graph of $f$. Then $G+G$ is compact: algebraic sum of a graph of continuous function and itself Borel or ...