Continuous exact frame is defined in the following paper. Since it is written by mathematicians, it should be suitable for you to study.
"A New Approach to Continuous Riesz Bases", A. A. Arefijamaal, R. A. K. Gol, R. R. Tousi and N. Tavallaei, Journal of Sciences, Islamic Republic of Iran, vol. 24(1), pp. 63-69, 2013.