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We present ab initio calculations of uniaxial absolute deformation potentials of the valence and the conduction bands in monolayer MoS2, MoSe2, WS2, WSe2, h-BN, and phosphorene. Calculations are performed using both semilocal and hybrid functionals. The absolute positions of the band edges in strained and unstrained materials are determined using the vacuum level as reference. For WSe2, we compare the obtained results with measured shifts of the valence band maximum (VBM) and the conduction band minimum (CBM) induced by uniaxial strain and find a very good agreement. The parameters describing the shifts in the VBM and CBM positions under strain can be used in the modeling of devices such as tunneling field-effect transistors.