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If f(t) belongs to L(0, R) for every positive R and is such that the integralconverges for x > 0, then F(s) exists for complex s(s ╪ 0) not lying on the negative real axis andfor any positive ξ at which f(ξ+) and f(ξ−) both exist.We define an operator Lk, t[F(x)]byUnder the above conditions on f(t), it is known that for all points t of the Lebesgue set for the function f(t),Let Ln, x denote the differentiation operatorSuppose thatconverges for some x¬ 0; then, if f(t) belongs to L(R−1, R) for every R>1,

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In this paper we define a continuous wavelet transform of a Schwartz tempered distribution $f \in S^{'}(\mathbb R^n)$ with wavelet kernel $\psi \in S(\mathbb R^n)$ and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of $S^{'}(\mathbb R^n)$. It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.