Geodesic
Curves with zero derivative in F-spaces
Glasgow mathematical journal, Tome 22 (1981) no. 1, pp. 19-29

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Let X be an F-space (complete metric linear space) and suppose g:[0, 1] → X is a continuous map. Suppose that g has zero derivative on [0, 1], i.e.for 0≤t≤1 (we take the left and right derivatives at the end points). Then, if X is locally convex or even if it merely possesses a separating family of continuous linear functionals, we can conclude that g is constant by using the Mean Value Theorem. If however X* = {0} then it may happen that g is not constant; for example, let X = Lp(0, 1) (0≤p≤1) and g(t) = l[0,t] (0≤t≤1) (the characteristic function of [0, t]). This example is due to Rolewicz [6], [7; p. 116]. 
Kalton, N. J. Curves with zero derivative in F-spaces. Glasgow mathematical journal, Tome 22 (1981) no. 1, pp. 19-29. doi: 10.1017/S0017089500004432
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