I don't have my original ID with me, but I do have an image of my ID in my mobile phone. Can I fly from Delhi to Kolkata?
You need an original ID. You could be prevented from travelling further, either at the entrance of the airport, or at check in, if you don't have an original document.
I am from India. If you are an Indian and have soft copy of Aadhar Card then you certainly travel anywhere with it. That will work as your ID proof, you will not require hard copy. But, if you are foreigner then hard, original ID is mandatory or else they will not let you exit. Also, you will have trouble while going back to your country.
Let me start by saying that I strongly recommend that you carry an original government issued ID proof with you while travelling by air. There is a wide list of documents you can carry including:
5-Voters Identity Card
6-Photo ID card issued by any of the following organizations/Institutions
Central Government or any of its Ministries.
Statutory / Regulatory authorities
State Govt. or any of its Ministries
Public Sector Undertaking (established under GOI or State Govt)
State Govt. of J&K
Senior Citizen card issued by State/Central Govt.
Govt. of India to Persons of Indian Origin [PIO card]
Defence Dept / Ministry of defence for Defence personnel & their dependents
Public Financial institutions / Public sector banks
But if you really want to take a chance then I can also tell you that I've successfully entered airports on Credit and Debit Cards with a picture