Floor function

Plugging in $1990$ for $x$ gets us:

$14=f(1990)-90\lfloor\frac{f(1990)}{90}\rfloor$

Let $f(1990)=90a+b$ for some integers $a$ and $b$ , with $0\leq b<90$ .

Then $90\lfloor\frac{f(1990)}{90}\rfloor=90a$ , and

$14=90a+b-90a=b$

So we are looking for all positive intgegers between $1900$ and $2000$ that are congruent to $14 \mod 90$ . These are $1904$ and $1994$ , so their sum is $\boxed{3898}$ .