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Here we study the systematic evolution of the topological properties of a Chern insulator in presence of an electronic dispersion that can be tuned smoothly from being Dirac-like till a semi-Dirac one and beyond. The band structure under such controlled deformation shows that the two Dirac points approach each other, merge at an intermediate point (the ${\mathbf{M}}$ point), where the low energy spectrum turns gapless, shows anisotropic Dirac features in the $k$-space and is denoted as the semi-Dirac limit, however a gap eventually opens up again in the spectrum. The Chern number phase diagram obtained via integrating the Berry curvature over the Brillouin zone (BZ) shows a gradual shrinking of the 'topological' lobes, and vanishes just beyond the semi-Dirac limit of the electronic dispersion. Thus there is a phase transition from a topological phase to a trivial phase across the semi-Dirac point. The vanishing of the anomalous Hall conductivity plateau and the merger of the chiral edge states with the bulk bands near the ${\mathbf{M}}$ point provide robust support of the observed phase transition.