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Contrary to numerous experimental evidences and conventional concepts, the existence of intrafunctional hydrogen bonds (IFHBs) in COOH groups of carboxylic acids and intramolecular HBs (IMHBs) between vicinal hydroxyl groups in 1,2-diols and other compounds is challenged in the last decade. Conclusions on nonexistence of such HBs are based on the lack of (3,-1) critical point (CP) between a hydrogen atom HD of a proton donor and an oxygen atom OA of a proton acceptor in electron density (ED) distributions in the molecules of such compounds. According to the quantum theory of atoms in molecules, the presence of a (3,-1) CP and its associated atomic interaction line between two atoms of a molecule in its stationary state is both necessary and sufficient to consider that the two atoms are bonded to one another. The delocalization indices DIs, the source function analysis, and the interacting quantum atom (IQA) approach are used to analyze the intrafunctional HD•••OA interactions in formic acid, conformers of lactic and succinic acids as well as HD•••OA interactions between vicinal hydroxyls in conformers of 1,2-ethanediol and 1,2,3-trihydroxypropene (the enol of 1,3-dihydroxyacetone). The MP2(FC)/cc-pVTZ and HF electron density distributions are used to calculate DIs(HD,OA) and to perform IQA analysis, respectively. DIs(HD,OA) as well as energies of electrostatic and exchange interactions between HD and OA atoms show that the intrafunctional hydrogen bonding interactions in carboxylic acids and IMHB between vicinal hydroxyl groups in 1,2-diol and 1,2,3-triol actually exist. However, atomic basins of C and OD atoms (OD is the oxygen atom of a hydrogen bond donating HD-OD group) and their EDs reach into the space between HD and OA atoms of COOH groups preventing the occurrence of (3,-1) CPs in ED distributions between these atoms in the acids. Also, the contributions of other atoms (particularly OD atom) to EDs between HD and OA atoms of five membered rings formed due to the formation of IMHBs in 1,2-diols prevent the occurrence of (3,-1) CPs in ED distributions between these atoms in these diols. The data obtained show that the presence of a (3,-1) CP and its associated bond path between two atoms of a molecule is thus not necessary for the existence of a bonding between these atoms as the lack of a minimum in ED distribution and, as a consequence, a BP between atoms in question may change but not “switch off” electrostatic, charge-transfer, exchange, correlation and other bonding interactions between the atoms.