1. Interpret the function carefully

The given function is

Also, the statement says $[x]$ denotes the greatest integer function, but it does not actually appear in the displayed formula. So we solve using the function as written.

We need all ordered pairs $(a,b)$ such that $f(x)$ is continuous at $x=3$.

2. Simplify the expression for $x\ne 3$

First factor:

Also,

Hence for $x\ne 3$,

So for $x\ne 3$,

where $\operatorname{sgn}(t)=\dfrac{t}{|t|}$ for $t\ne 0$.

3. Check the sign near $x=3$

Near $x=3$,

• If $x\to 3^-$, then $x-3<0$ and $x-4<0$, so

  $$(x-3)(x-4)>0$$

  Therefore,

  $$f(x)= -\frac{a}{b} \quad (x<3 \text{ near } 3)$$

• If $x\to 3^+$, then $x-3>0$ and $x-4<0$, so

  $$(x-3)(x-4)<0$$

  Therefore,

  $$f(x)= +\frac{a}{b} \quad (x>3 \text{ near } 3)$$

Thus,

4. Condition for continuity at $x=3$

For continuity at $x=3$, the left-hand and right-hand limits must be equal, and equal to $f(3)=b$.

So first,

This gives

Since the expression for $x\ne 3$ has denominator $b$, we must have

Hence,

Then both one-sided limits become $0$. For continuity, we also need

So this requires

But this contradicts the requirement $b\ne 0$ for the formula when $x\ne 3$ to be defined.

Therefore, no ordered pair $(a,b)$ exists such that $f(x)$ is continuous at $x=3$.

So,

and the number of elements is

5. Compare with the options

The correct count should be $0$, but this is not among the options.

So the stored answer $\boxed{\text{D: }1}$ does not agree with the mathematics of the displayed function.

It is likely there is a typo in the question statement or formula.

6. Final conclusion

The number of ordered pairs $(a,b)$ is

Hence the stored answer $D$ is not correct for the function as written.