Orthonormal family in L^2

Suppose f in L^1([0,1])\L^2([0,1]).
Prove that there exists a complete orthonormal basis phi_n for L^2([0,1]) such that for each n,phi_n is continuous and moreover int_0^1 f(x)phi_n(x)dx = 0


Can we do the following here : Just take any coutable basis of L^2([0,1]),consisting of continuous functions say phi_n = x^n. Then consider the countabke family f, phi_1, phi_2,...,phi_n,... and ortghogonalize it using Gramm - Schmidt process.

Thanks.
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