Meet-continuous ($\wedge$-continuous) element of a lattice $L$ is an element $a$ which satisfies \[ a\wedge\bigvee_{iı I}x_i=\bigvee_{iı I}(a\wedge x_i) \] for every chain $\{x_i\mid i\in I\}$ of a lattice; an element with the dual property is called join-continuous ($\wedge$-continuous), and element with both properties - continuous. We give some properties of these elements, and prove some statements on lattice identities containing infinitary operations. In the second part we apply these lattice theoretic results on the lattice of weak congruences of an algebra. Namely, the diagonal relation of this lattice is always a meet-continuous element in the lattice of weak congruences. We consider some well known algebraic properties, such as CEP, CIP and infinite CIP ($^*$CIP) and their connection with the continuity of the diagonal relation. Particularly, we prove some results on transferring these properties from an algebra to it's subalgebras or factor algebras.