Select numbers $a$ and $b$ between 0 and 1 independently and at random, and let $c$ be their sum. Let $A$, $B$, and $C$ be the results when $a$, $b$, and $c$, respectively, are rounded to the nearest integer.  What is the probability that $A+B=C$?
The conditions under which $A+B=C$ are as follows.


(i) If $a+b< 1/2$, then $A=B=C=0$.


(ii) If $a\geq 1/2$ and $b<1/2$, then $B=0$ and $A=C=1$.


(iii) If $a<1/2$ and $b\geq 1/2$, then $A=0$ and $B=C=1$.


(iv) If $a+b\geq 3/2$, then $A=B=1$ and $C=2$.

These conditions correspond to the shaded regions of the graph shown. The combined area of those regions is 3/4, and the area of the entire square is 1, so the requested probability is $\boxed{\frac{3}{4}}$.

[asy]
unitsize(2cm);
draw((1.1,0)--(0,0)--(0,1.1),linewidth(1));
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,gray(0.7));
fill((0.5,0)--(0.5,0.5)--(0,0.5)--cycle,white);
fill((0.5,0.5)--(1,0.5)--(0.5,1)--cycle,white);
label("$a$",(1.1,0),E);
label("$b$",(0,1.1),N);
label("1",(1,0),S);
label("1",(0,1),W);
label("0",(0,0),SW);
[/asy]