Flett's mean value theorem in topological vector spaces (Q5957251)

Starting point is a result of \textit{T. M. Flett} [Math. Gaz. 42, 38-39 (1958)]. Namely that for every differentiable real valued function \(f\) on an interval \([a,b]\) with identical derivative at the endpoints there exists a point \(\eta\in[a,b]\) such that the tangent of \(f\) at \(\eta\) passes through the endpoint \((a,f(a))\). NEWLINENEWLINENEWLINEHere it is shown that this can be adapted to situations where the assumption on the endpoints is not satisfied or the differentiability assumption is weakened. Furthermore a weak version of this for topological vector space valued functions is given.