Geodesic
Peano's theorem in an infinite-dimensional Hilbert space is false even in a~weakened formulation
Matematičeskie zametki, Tome 15 (1974) no. 3, pp. 467-477

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We formulate a continuous function $F\colon R\times H\to H$, where $H$ is a separable Hilbert space such that the Cauchy problem $$ x'(t)=F(t,x(t)),\quad x(t_0)=x_0 $$ has no solution in any neighborhood of the point $t_0$, no matter what $t_0\in R$ and $x_0\in H$ are considered. 
@article{MZM_1974_15_3_a13,
     author = {A. N. Godunov},
     title = {Peano's theorem in an infinite-dimensional {Hilbert} space is false even in a~weakened formulation},
     journal = {Matemati\v{c}eskie zametki},
     pages = {467--477},
     publisher = {mathdoc},
     volume = {15},
     number = {3},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_15_3_a13/}
}
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A. N. Godunov. Peano's theorem in an infinite-dimensional Hilbert space is false even in a~weakened formulation. Matematičeskie zametki, Tome 15 (1974) no. 3, pp. 467-477. http://geodesic.mathdoc.fr/item/MZM_1974_15_3_a13/