We give an elementary proof of the following theorem of Titchmarsh. Suppose f , g f,g are integrable on the interval ( 0 , 2 T ) \left ( {0,2T} \right ) and that the convolution f ∗ g ( t ) = ∫ 0 t f ( t − x ) g ( x ) d x = 0 f * g\left ( t \right ) = \int _0^t {f\left ( {t - x} \right )g\left ( x \right )dx} = 0 on ( 0 , 2 T ) \left ( {0,2T} \right ) . Then there are nonnegative numbers α , β \alpha ,\beta with α + β ≥ 2 T \alpha + \beta \geq 2T for which f ( x ) = 0 f\left ( x \right ) = 0 for almost all x x in ( 0 , α ) \left ( {0,\alpha } \right ) and g ( x ) = 0 g\left ( x \right ) = 0 for almost all x x in ( 0 , β ) \left ( {0,\beta } \right ) .