Proving boundedness of a function (part 1).

Question

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; \forall 1 \leq i \leq n,\; \sum_{i}^{n}b_ix_i \neq 0 \}$. Assume that the constants $(a_i, b_i), 1 \leq i \leq n$ are all strictly positive. Is the function $f(x)$ bounded over the set $S$.

Answer 1

It is not bounded. Take for example $n=2$, $a_1=a_2=1$ and $b_1=1,b_2=1/2$. Then we can make the denominator arbitrarily closed to zero by taking $x_2=1$, and saying $x_1$ getting close to $-1/2$. Then, the numerator won't be zero, but the denominator will converge to zero making the function diverge to infinity.