In this paper we continue the research in Real Analysis, in the spirit of the very recent book . Firstly, as a refinement of the global defect of integrability introduced in , $\S $5.1, we consider here a pointwise defect of integrability and study its properties and connections with the pointwise defect of continuity already introduced in , $\S$ 5.1. Secondly, as refinements of the global defects of monotonicity and of convexity introduced in the same book , $\S$ 5.2, we consider and study pointwise variants
"On some pointwise defects of properties in real analysis.." Real Anal. Exchange 29 (1) 175 - 197, 2003-2004.