Continuity of integral of norm of Jacobian matrix

Question

I have a question about a step in the proof of Theorem 1 of Section 2.8 of Sontag, Mathematical Control Theory (a pdf is available from the author here). The statement of the theorem can be found in page 57, but my problem is in the last part of the proof, in page 60. There we have a function $f: \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$ which is of class $C^1$ and of compact support. The author writes (subscripts indicate partial Jacobians) $$ f(x + a, u + b) - f(x, u) - f_x(x, u)a - f_u(x, u)b = N(x, u, a, b) $$ where $$ N(x, u, a, b) = \int_0^1{[(f_x(x+ta, u+tb) - f_x(x, u))a + (f_u(x+ta, u+tb) - f_u(x, u))b]dt}. $$ In particular, N is jointly continuous in its arguments and thus uniformly continuous since it has compact support. I have no problem understanding this. What escapes me is what is claimed next, that the function $C$ defined by $$ C(a, b) = \sup_{(x, u) \in \mathbb{R}^n \times \mathbb{R}^m} \int_0^1{|f_x(x+ta, u+tb) - f_x(x, u)| + |f_u(x+ta, u+tb) - f_u(x, u)| dt} $$ is continuous. Here $|\cdot|$ denotes the matrix norm induced from the appropriate Euclidian norms. I am having a hard time filling in the missing steps; I can see that the supremum over $(x,u)$ of $N$ is continuous, but cannot figure out how to translate that to continuity of $C$.