This is the lowest closing for the domestic currency since April 18.
The relatively strong rupee once again turned shaky and wobbled after a brief overnight recovery under immense dollar pressure on mounting expectations that the Federal Reserve would raise interest rates next month.
The rupee resumed substantially lower at 64.42 against the Monday's closing of 64.31 at the Interbank Foreign Exchange market on fresh bouts of dollar demand.
The RBI, meanwhile, fixed the reference rate for the dollar at 64.5065 and for the euro at 70.4669.
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Currency traders also exercised an abundance of caution ahead of a forex market holiday and also speeches by key FOMC members later in the day.
The forex market will remain closed tomorrow on account of 'Buddha Purnima'.
Heavy capital outflows also impacted the sentiment as foreign funds sold shares worth a net Rs 542.47 crore yesterday, as per the provisional figures from stock exchanges.
The flagship Sensex edged higher by 7.10 points to end at 29,933.25 after briefly climbing the 30,000-mark in early trade. The broader Nifty was up 2.80 points at 9,316.85.
In worldwide trade, the US dollar traded firmly highest against major trading rivals as currency markets swung back to trading on the diverging outlooks for growth and tighter monetary policy on Tuesday, lifting the dollar and euro against the yen.
Emmanuel Macron's election on Sunday put a cap on immediate political concerns, pushing US and European yields higher.
In cross-currency trade, the rupee declined further against the pound sterling to finish at 83.52 from 83.33.
However, the rupee firmed up further against the euro to settle at 70.38 as compared to 70.45 and also rose against the Japanese Yen to close at 56.79 per 100 yens from 57.11 yesterday.
In the forward market today, premium for dollar continued to rule subdued owing to sustained receivings from exporters.
The benchmark six-month premium for October edged down to 150.5-151.5 paise from 151-153 paise and the far-forward April 2018 also slipped to 307-309 paise from 309-311 paise earlier.
Brent crude futures were up 20 cents at USD 49.54 per barrel in early Asian trade.
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