The rupee fell by 19 paise to 45 rupees 40 paise per US dollar on Wednesday. This follows renewed demand for the American currency from banks and importers amid strengthening of the dollar in overseas markets and weakness in the domestic equity market. At the Interbank Forex Exchange, the rupee resumed lower at 45 rupees 29 per dollar as against the previous closing level. The dollar rose against its European counterpart today, benefiting from persistent worries about euro-zone debt problems.