Fourier transform of a function with reciprocal tails

Suppose $w$ is locally integrable andThen $\hat{w}(\lambda )+(c_{+}-c_{-})\log |\lambda |$ has finite limits at zero from both sides. With the convention $\hat{w}(\lambda )=\int e^{-i\lambda x}w(x)dx$, the positive-frequency limit minus the negative-frequency limit isSubtracting suitable reflected copies of $x^{-1}{1}_{(1,\infty )}(x)$ leaves an $L^1$ function, whose Fourier transform is continuous; the jump then follows from the Dirichlet integral.