Histone demethylase that specifically demethylates 'Lys-9' and 'Lys-36' residues of histone H3, thereby playing a central role in histone code (PubMed:16603238, PubMed:26741168, PubMed:21768309). Does not demethylate histone H3 'Lys-4', H3 'Lys-27' nor H4 'Lys-20' (PubMed:16603238, PubMed:26741168, PubMed:21768309). Demethylates trimethylated H3 'Lys-9' and H3 'Lys-36' residue, while it has no activity on mono- and dimethylated residues (PubMed:16603238, PubMed:26741168, PubMed:21768309). Demethylation of Lys residue generates formaldehyde and succinate (PubMed:16603238). Also able to demethylate histone H1-4 methylated at 'Lys-26' (H1.4K26me1, H1.4K26me2 and H1.4K26me3) (PubMed:19144645, PubMed:30156264). Participates in transcriptional repression of ASCL2 and E2F-responsive promoters via the recruitment of histone deacetylases and NCOR1, respectively (PubMed:16024779). {ECO:0000269|PubMed:16024779, ECO:0000269|PubMed:16603238, ECO:0000269|PubMed:19144645, ECO:0000269|PubMed:21768309, ECO:0000269|PubMed:26741168, ECO:0000269|PubMed:30156264}. [Isoform 2]: Crucial for muscle differentiation, promotes transcriptional activation of the Myog gene by directing the removal of repressive chromatin marks at its promoter. Lacks the N-terminal demethylase domain. . This is the function of Ensembl gene identifier ENSG00000066135 (KDM4A, lysine demethylase 4A).