Generalized essential variation and quasi-continuity

Summary

For functions f : E →Y, where E ˆ R is a semi-open set and Y is a metric space, the essential φ-variation V_φ,℘ (f, E) depending on some σ-ideal in R is defined. In the case where f is quasi-continuous, V_{φ, ℘} (f, E) is equal to the φ-variation V_φ (f, E). If in addition Y is complete, then each function f of bounded φ -variation is graph quasi-continuous.