Unusual situations that arise with the Dirac delta function and its derivative

There is a situation such that, when a function ƒ() is combined with the Dirac delta function δ(), the usual formula does not hold. A similar situation may also be encountered with the derivative of the delta function δ'(), regarding the validity of . We present an overview of such unusual situations and elucidate their underlying mechanisms. We discuss implications of the situations regarding the transmission-reflection problem of one-dimensional quantum mechanics.