If $f(x)$ is a bounded function on $a,b,$ then $f(x)$ is integrable on $a,b$ if and only if

a$)\frac{?}{\text{b}\text{a}\text{r}}({\displaystyle \underset{a}{\overset{b}{}}})f(x)dx={\displaystyle \underset{a}{\overset{b}{}}}{?}_f(x)dx.$

b$)$ for every $lon>0$ there is a partition $P$ of $a,b$ with

c$)$ for every $lon>0$ there exists a $>0$ such that if $P$ and $Q$ are partitions of $a,b$ such that $||P||<mi></mi>$ and then

$|S(f,P)-S(f,Q)|<mi></mi><mi>l</mi><mi>o</mi><mi>n</mi>$

for any Riemann sums $S(f,P)$ and $S(f,Q).$

d$)$ All of the above.