If we assume that $f$ and $g$ are two continuous functions such that $f_{|\gamma}=g_{|\gamma}$ where $\gamma$ is the interface between two triangles $K1$ and $K2$ (we are in the context of finite elements), we can write obviously:
$f_{|\gamma}=g_{|\gamma}$ in the space $C^{0}(\gamma)$. (1)
Now, assume I do not have the equality above, but the following equality:
$\int_\gamma f_{|\gamma}\varphi_{\sigma}=\int_\gamma g_{|\gamma}\varphi_{\sigma}$. (2)
for all $\sigma$ where $\varphi_\sigma$ is the Lagrange basis function associated to the degree of freedom $\sigma$ defined for the finite element $(\gamma,P^k,\Sigma_\gamma)$ (again, I am using the standard notations of the finite element methods).
I think I can say that:
(2) is a weaker statement than (1),
(2) is true for all polynomials $p$ of $P^k(\gamma)$ by linearity: $\int_\gamma fp=\int_\gamma gp$.
How can I interpret the equality (2) ?