Continuing its relentless fall, the rupee touched a new low against the dollar, as it breached the 56-mark against the US currency in afternoon trade, at the foreign exchange market, today. The rupee fell on persistent dollar demand from banks and importers, amid a weak domestic stock market. Earlier this morning, the rupee had resumed weaker, at 55.65 per dollar, compared to yesterday's closing of 55.39 per dollar.The dollar has been gaining strength against other global currencies as well. In the New York market, the dollar rose to its highest level since January against a basket of major currencies, after the Japanese yen dropped, following rating agency, Fitch cutting Japan's sovereign rating to A plus, from AA.