What are the properties of indirect utility function?

Properties of the indirect utility function:

While u(x) is continuous and monotonic onto R^L₊ and (p, m) > 0, the indirect utility function has the given properties as follows:

a. v(p, m) is nonincreasing in p; which is, when p′> p, v(p′, m) < v(p, m). Likewise, v(p, m) is nondecreasing in m.

b. v(p, m) is homogeneous of degree 0 into (p, m).

c. v(p, m) is quasiconvex in p; which is, {p: v(p, m) < k} is a convex set for each k.

d. v(p, m) is continuous at all p >> 0, m > 0.