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Let X be a completely regular Hausdorff topological space and let C(X) (the set of all real‐valued bounded and continuous in X functions) be endowed with the sup‐norm. Let ßX, as usual, denotes the Stone‐Čech compactification of X. We give a characterization of those X for which the set left{f ∈ C(X): the extension of on βX attains its maximum on βXonly at point of X} contains a dense scriptGδ‐subset of C(X). These are just the spaces X which contain a dense Čech complete subspace. We call such spaces almost Čech complete. We also prove that X contains a dense completely metrizable subspace, if, and only if, C(X) contains a dense scriptGδ‐subset of functions which determine Tykhonov well‐posed optimization problems over X. For a compact Hausdorff topological space X the latter result was proved by Čoban and Kenderov [CK1.CK2]. Relations between the well‐posedness and Gâteaux and Fréchet differentiability of convex functionals in C(X) are investigated. In particular it is shown that the sup‐norm in C(X) is Frechet differentiable at the points of a dense scriptGδ‐subset of C(X), if, and only if, the set of isolated points of X is dense in X. Conditions and examples are given when the set of points of Gateaux differentiability of the sup‐norm in C(X) is a dense and Baire subspace of C(X) but does not contain a dense scriptGδ‐subset of C(X).

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