{I/(

Theorem 13.10. Suppose that the junction c is jointly continuous, and the junctions I( Ip, " " '), m(Ip, " "'), and n(Ip, " ',', .) are Ll-continuous and L1-bounded for each Ip. Then the operator junction P(·) is norm continuous in the space £(C). o First of all, the continuity of the function c and the Lt-continuity and Lt-boundedness of the functions I, m and n at I() are obviously sufficient for the operator P(I() to act in C(D). Given Ipo E J, we show that lim I\P(Ip) - P(lpo)1\ = O. We have

IIP(Ip) - P(l()o)1\ = sup II[P(Ip) - P(l()o)]zll 1I%1I~1

= sup max /[P(Ip) - P(lpo)]z(t,s)1 1I%1I~1 (t,.)ED

Since the function c(', t, s) is continuous at I()o and the functions I, m and n are Lrcontinuous, for each E > 0 we can find a 6 > 0 such that