I don't get this Rudin PMA, 245-246 statement:
'Suppose $I^k$ is $k$-cell in $\Bbb R^k$, consisting of all $\vec {x}=(x_1,\ldots,x_k)$ such that $$a_i\le x_i\le b_i\,\,\,(i=1,\ldots,k)\tag 1$$. $I^j$ is $j$-cell in $\Bbb R^j$ defined by the first $j$ inequalities (1). $f$ real continuous function on $I^k$. Put $f=f_k$ and define $f_{k-1}$ on $I^{k-1}$ by $$f_{k-1}(x_1,\ldots,x_{k-1})=\int_{a_k}^{b_k}f_k(x_1,\ldots,x_{k-1},x_k)\,dx_k.$$ The uniform continuity of $f_k$ on $I_k$ shows that $f_{k-1}$ is continuous on $I^{k-1}$.' Why is $f_{k-1}$ continuous?