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Is [0,1] Closed at the Origin?

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Please solve the following problem:

Let C ([0,1]) be the space of continuous functions in [0,1] with the norm
II f II = max I f(x) I on [0,1].
Is the subspace of functions that are = 0 at the origin closed subspace in
C^0 [o,1] with this norm/Prove or disapprove.

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Solution Summary

The solution proves or disapproves if [0,1] closed at the origin.

Solution Preview

Yes, it is closed. Suppose, {f_n} is a sequence of functions from the subspace Y={g in C([0,1]): g(0)=0} converging to some function f in C([0,1]) in ...

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