What is optimal choice of consumer according to consumer behavior?

Consumer's Optimal Choice:

In the fundamental problem of preference maximization, the set of affordable options is just the set of all bundles which satisfy the consumer's budget constraint. It is the problem of preference maximization can be illustrated as:

max u(x)

as like  px < m

x is in X.

There will existence a solution to such problem when the utility function is continuous and which the constraint set is bounded and closed.

The constraint set is exactly closed. When p_i > 0 for i = 1, ..., k and m > 0, this is not complicated to demonstrate that the constraint set will be bounded. When some price is zero, the consumer might wish for an infinite amount of the equivalent good.