Final answer:
The analyticity of f(z) in a region R implies that its component function v(x, y) must be continuous and have continuous partial derivatives, except at points where it could be unbounded, which is not suggested by the given conditions.
Step-by-step explanation:
The question is asking to show that the component function v(x, y) of a complex function f(z) = u(x, y) + iv(x, y) behaves in a certain way given that f(z) is continuous on a closed bounded region R, is analytic and not constant in the interior of R. Analyticity implies that both u(x, y) and v(x, y) satisfy the Cauchy-Riemann equations and have continuous partial derivatives.
Since the function f(z) is continuous and analytic in R, and given the relationship between the analyticity of a complex function and its component functions, one can infer that v(x, y) is continuous in R. Additionally, because f(z) is not constant, v(x, y) also exhibits non-constant behavior within R. According to the information provided, this implies that the derivative of v(x, y) with respect to space, which we could write as ∂v/∂x and ∂v/∂y (since v is a function of both x and y), must be continuous unless the function becomes unbounded, which is not indicated by the premise.