Consider the Banach space C[0,1] of continuous functions on the interval [0,1] equipped with the sup-norm. Let T: C[0,1] -> C[0,1] be a bounded linear operator such that T(f) is continuously differentiable for every f in C[0,1]. Prove or disprove the following statement: "If T is injective, then T^{-1} is also bounded."

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