Yes, you can buy and sell stocks across the exchanges. This is facilitated by interoperability among clearing corporations of different exchanges. This means that trades made on any exchange can be settled through a single clearing corporation.
Let's say you buy ITC on BSE for Rs.200. You can sell it on NSE at Rs.202 for a profit of Rs.2.
Please note that the above example does not include Brokerage and other regulatory fees levied by your broker. Your actual profit will be net of such costs for both Buy and Sell.
You can read more about it in this SEBI circular.
Yes, you can buy and sell stocks across the exchanges.