In previous work, we have considered the problem of showing that a continuous function on a real hypersurface $\Gamma$ in $\C^N$ satisfies the tangential Cauchy-Riemann equations provided that its slices satisfy conditions of Morera type. For instance, these results imply that if $\Omega \subset \C^N$ is a bounded convex domain with smooth boundary, strictly convex at $z_0\in bD$, if $L_0$ is a complex line tangent to $b\Omega$ at $z_0$ and if $f$ is a continuous function on $b\Omega $ such that $\int _{L\cap b\Omega} f\omega = 0$ for all complex lines $L$ close to $L_0$ which meet $\Omega $ and for all $(1,0)$ forms with constant coefficients, then $f$ is a CR function in a neighbourhood of $z_0$. This fails to hold if $L_0$ is a complex line that meets $\Omega$ even under much stronger assumption of holomorphic extendibility along complex lines. Indeed, let $\mathbb B$ be the open unit ball in $\C ^2$, and define a function $f$ on $b\mathbb B \setminus \{z=0\}$ by $f(z,w) = 1/\overline z $. It is easy to verify that for each complex line $L$ close to the $z$-axis, \ $f|L\cap b\mathbb B$ has a continuous extension to $L\cap \overline {\mathbb B }$ which is holomorphic on $L\cap \mathbb B $, yet there is no open set in $b \mathbb B $ on which $f$ is a CR function. So to conclude that $f$ is a CR function one has to assume the holomorphic extension property for a larger family of analytic discs.