The rupee rebounded from its all-time low level and inched higher by 2 paise to 78.02 against the US dollar in opening trade on Tuesday. At the interbank foreign exchange, the rupee opened at 78.02 against the dollar, registering a rise of 2 paise over its previous close. In initial deals, the domestic unit also surged to 77.90 against the American currency but could not hold on to the gains and touched 78.06. On Monday, the rupee depreciated by 11 paise to close at a fresh lifetime low of 78.04 against the US dollar.