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Mathematicians also study weak solutions (relying on weak derivatives), which are types of solutions that do not have to be differentiable everywhere. en.wikipedia.org

Also, monsters (nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves) began to be investigated. en.wikipedia.org

A function is (totally) differentiable if its total derivative exists at every point in its domain. en.wikipedia.org

A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval. en.wikipedia.org

For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly. en.wikipedia.org

Since differentiable manifolds are topological spaces we know what it means for "f" to be continuous. en.wikipedia.org

First, it implies that a function which is holomorphic in an open set is in fact infinitely differentiable there. en.wikipedia.org

In geometry and topology, all manifolds are topological manifolds, possibly with additional structure, most often a differentiable structure. en.wikipedia.org

If the derivative exists everywhere, the function is differentiable. en.wikipedia.org

So differentiable structures on a manifold is an example of topology. en.wikipedia.org

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