Remark 2

To show that a continuous function admits a primitive, one can use the fact that every polynomial function of a real variable (with coefficients in $E$) admits a primitive; since from the theorem of Weierstrass (Gen. Top., X, p. 313, prop. 3) every continuous function is the uniform limit of polynomials on every compact interval, th. 1 of II, p. 52 shows that every continuous function admits a primitive.