<!-- The 19th century saw great advances in the theory of [[real analysis]], including theories of convergence of functions and [[Fourier series]]. Mathematicians such as [[Karl Weierstrass]] began to construct functions that stretched intuition, such as [[Continuous, nowhere differentiable function|nowhere-differentiable continuous functions]]. Previous conceptions of a function as a rule for computation, or a smooth graph, were no longer adequate.  Weierstrass began to advocate the [[arithmetization of analysis]], which sought to axiomatize analysis using properties of the natural numbers. The modern [[(ε, δ)-definition of limit]] and [[continuous function]]s was already developed by [[Bernard Bolzano|Bolzano]] in 1817 ([[#CITEREFFelscher2000|Felscher 2000]]), but remained relatively unknown.
[[Cauchy]] in 1821 defined continuity in terms of [[infinitesimal]]s (see Cours d'Analyse, page 34).  In 1858, Dedekind proposed a definition of the real numbers in terms of [[Dedekind cuts]] of rational numbers [[#CITEREFDedekind1872|(Dedekind 1872)]], a definition still employed in contemporary texts.
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