The limit of convergent continuous function sequencemay be discontinuous.This makes difficults both in theory and applica-tion.For example,it makes the condition for operation of limits of integralof the function sequence become very strong.In general,it requiresuniform convergence to the function sequence.Therefore it is very imp-ortant to study such convergence as the limit of continuous function sequence,the limit is continuous.This paper studies a known substrong converg-ence of a function sequence in topological space and proves the necessaryand sufficient conditionof this convergence.