Prove or disprove the existence of a function \( f:[0, 1] \to [0, 1] \) with the following property:
for any interval \( (a, b) \subset [0, 1] \) with \( a<b \), \( f((a, b)) = [0, 1] \).
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Prove or disprove the existence of a function \( f:[0, 1] \to [0, 1] \) with the following property:
for any interval \( (a, b) \subset [0, 1] \) with \( a<b \), \( f((a, b)) = [0, 1] \).