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The aim of this expository article is to present recent developments in thecenturies old discussion on the interrelations between continuous anddifferentiable real valued functions of one real variable. The truly newresults include, among others, the $D^n$-$C^n$ interpolation theorem: {\em Forevery $n$-times differentiable $f\colon\R\to\R$ and perfect $P\subset \R$ thereis a $C^n$ function $g\colon\R\to\R$ such that $f\restriction P$ and$g\restriction P$ agree on an uncountable set} and an example of adifferentiable function $F\colon\R\to\R$ (which can be nowhere monotone) and ofcompact perfect $\mathfrak{X}\subset\R$ such that $F'(x)=0$ for all $x\in\mathfrak{X}$ while $F[\mathfrak{X}]=\mathfrak{X}$; thus, the map$\mathfrak{f}=F\restriction\mathfrak{X}$ is shrinking at every point while,paradoxically, not globally. We also present a new short and elementaryconstruction of {\em everywhere differentiable nowhere monotone $h\colon\R\to\R$}\/ and the proofs (not involving Lebesgue measure/integration theory)of the theorems of Jarn\'\i k and of Laczkovich. The main part of thisexposition, concerning continuity and first order differentiation, is presentedin an narrative that answers two classical questions: \textit{To what extend acontinuous function must be differentiable?} and \textit{How strong is theassumption of differentiability of a continuous function?}  In addition, we overview the results concerning higher order differentiation.  This includes the Whitney extension theorem and the higher orderinterpolation theorems related to Ulam-Zahorski problem. Finally, we discussthe results concerning smooth functions that are independent of the standardaxioms ZFC of set theory. We close with a list of currently open problemsrelated to this subject.

Differentiability versus continuity: Restriction and extension theorems and monstrous examples