A partition P of interval \([a,b]\) is a finite set of points \(x_0,x_1,\cdots,x_n\), where \(a=x_0\le x_1\le \cdots\le x_n=b\) and \(\Delta x_i=x_i-x_{i-1}\quad(i=1,\cdots,n)\). Let \[M_i=\text{sup }f(x)\quad(x_{i-1}\le x\le x_i)\] \[m_i=\text{inf }f(x)\quad(x_{i-1}\le x\le x_i)\] Corresponding to each partition \(P\) of \([a,b]\). We put \[U(P,f)=\sum_{i=1}^{n}M_i\Delta x_i\] \[L(P,f)=\sum_{i=1}^{n}m_i\Delta x_i\] \[\overline{\int}_{a}^{b} f dx=\text{inf}\sum_{i=1}^{n}M_i\Delta x_i\] \[\underline{\int}_{a}^{b} f dx=\text{sup}\sum_{i=1}^{n}m_i\Delta x_i\] which are called upper and lower Riemann integrals of \(f\) over \([a,b]\), respectively. If the upper and lower integrals are equal \[\overline{\int}_{a}^{b} f dx=\underline{\int}_{a}^{b} f dx\] we say that \(f\) is Riemann-integrable on \([a,b]\), we write \(f\in\mathscr R\).
The case \(k=m=1\) is the fundamental theorem of calculus: If \(f\in \mathscr R\) on \([a,b]\) and if there is a differentiable function \(F\) on \([a,b]\) such that \(F'=f\), then \[\int_{a}^{b}f(x)dx=F(b)-F(a)\] Let \(\varepsilon>0\), choose a partition \(P=\{x_0,\cdots,x_n\}\) of \([a,b]\) so that \[U(P,f)-L(P,f)<\varepsilon\] let point \(t_i\in[x_{i-1},x_i]\) such that \[F(x_i)-F(x_{i-1})=f(t_i)\Delta x_i\] for \(i=1,\cdots,n\) Thus \[\sum_{i=1}^{n}f(t_i)\Delta x_i=F(b)-F(a)\] Then \[\Biggl|F(b)-F(a)-\int_{a}^{b}f(x)dx\Biggr|\leq \Biggl|F(b)-F(a)-\sum_{i=1}^{n}f(t_i)\Delta x_i\Biggr|+\Biggl|\sum_{i=1}^{n}f(t_i)\Delta x_i-\int_{a}^{b}f(x)dx\Biggr|<0+\varepsilon=\varepsilon\] Then then \[\int_{a}^{b}f(x)dx=F(b)-F(a)\]