Can I invest in mutual funds without having a source of income, i.e. from my savings?

Of course, you can invest from your savings. Mutual fund laws require that the money come from your own bank account. This is to prevent money laundering, in other words the source of money in your bank has to be from savings, past income, gift, inheritance etc.

When you invest, ensure you only do in Direct Plan Mutual Funds. You then get upto 40% more in long term portfolio size by avoiding commissions and brokers.