The additional function vested with the Collector by the East India Company was Judge Magistrate.

Explanation:
During the British rule in India, the East India Company established administrative units known as Collectorate. The Collector was the head of the Collectorate and had various functions and responsibilities. One of the additional functions vested with the Collector was that of a Judge Magistrate.

1. Collector's Role:
The Collector was responsible for the collection of revenue, maintenance of law and order, and administration of justice in the district. They were appointed by the East India Company and had extensive powers and authority.

2. Judge Magistrate Function:
As a Judge Magistrate, the Collector had the power to adjudicate and decide on civil and criminal cases within their jurisdiction. They acted as both a judge and a magistrate, ensuring the dispensation of justice in the district.

3. Judicial Powers:
The Collector, as a Judge Magistrate, had the authority to hear and decide civil disputes, criminal cases, and conduct trials. They had the power to impose fines, grant punishments, and pass judgments. Their decisions were binding and had legal validity.

4. Integration of Revenue and Judicial Functions:
The East India Company vested the additional function of a Judge Magistrate with the Collector to ensure the integration of revenue collection and judicial administration. This helped in maintaining control over the district, as the Collector had the power to both collect revenue and dispense justice.

5. Criticisms and Reforms:
The system of vesting judicial powers with the Collector was criticized for the potential conflict of interest and abuse of power. Over time, the British administration recognized the need for a separate judicial system and introduced reforms, leading to the establishment of a clear distinction between revenue and judicial functions.

In conclusion, during the British rule in India, the Collector appointed by the East India Company had the additional function of a Judge Magistrate. This allowed them to exercise judicial powers and administer justice within their jurisdiction. However, this system faced criticisms and was later reformed to establish a separate judicial system.