Precompact dots

Let M, N be metric spaces such that M is connected and N is complete and locally compact. Let \mathcal{F} be a uniformly equicontinuous family of functions from M to N such that there exists x_0\in M with the property (\star) that \mathcal{F}(x_0) is precompact. Prove that all points x\in M have property (\star).

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Precompact dots