In the field of irrigation engineering, a head regulator is an essential component in hydraulic structures such as barrages and dams. Its primary function is to manage and control the flow of water from the main river into an off taking canal, ensuring a regulated supply of water for irrigation and other purposes.

Let's evaluate the options to determine the correct function of a head regulator in a barrage:

1. To prevent erosion of riverbanks: This is not the primary function of a head regulator. Erosion control is generally achieved through other means such as protection works like revetments and spur dikes.
2. To prevent the river from outflanking the barrage: Preventing the river from bypassing the barrage is usually managed by training works or guide banks, rather than the head regulator itself.
3. To admit water into the off taking canal: This is the correct answer. The head regulator controls the volume of water that enters the canal from the barrage, ensuring an adequate and consistent water supply for irrigation systems beyond the canal.
4. To serve as one side wall of a fish ladder: Fish ladders are designed to help fish navigate around barriers like dams, but this is not a function of the head regulator.

Therefore, the main function of a head regulator is indeed to admit water into the off taking canal, making option (c) the correct choice.