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In ancient times, India was divided into smaller kingdoms like Kalinga, Magadh, Gandhar and so on. But most of these kingdoms followed the Aryan culture, Vedic rituals and religion and similar political set ups. You could say they were all part of one civilization. That is why, the concept of India was always there- from the time of Alexender's invasion to invasions by Sakas, Huns, Pathans, Mughals- all of them invaded 'India'. The travellers like Hieun Tsang, Fa Huen, Ibn Batuta talked about their visits to India. And that is why, while we do agree that it was the British who gave the final shape to what we today know as India, much before they came- Christopher Columbus had set out to 'discover' India. The British, French, Portuguese and Dutch all came to the land they had already known to be India because they had heard about it's riches. And from ancient times, many kings have tried to be emperor of this whole land that together formed India- Samudragupta, Kanishka, later the Mughals and finally the Europians. King Harshavardhana of Kanauj could not conquer the Southern part of India, so he called himself 'Uttarapathanatha'- the lord of the North. The title itself shows he acknowledged the un-vanquished South to be part of the same country which he could not conquer. 
Coming to the ancient name of this country- the ancient kings of the land called it Bharat after the name of King Bharat who had conquered the whole land. By ancient kings I mean the Aryans. Now the Aryans themselves were foreigners who settled by the river Sindhu. It is through mispronunciation that Sindhu became 'Hindu' or 'Indus' (Hinduism is therefore rather a way if life of the people who lived by the Sindhu than a religion, at least, that was what it was originally) and from there we have the names 'Hindusthan' and 'India.'