Explanation:

Introduction:
In complex analysis, an analytic function is a function that is locally given by a convergent power series. It is differentiable at every point in its domain. The real part of an analytic function is the real-valued function obtained by taking the real part of the complex-valued function.

Given:
We are given that f(z) is an analytic function whose real part is constant.

Real Part of an Analytic Function:
The real part of a complex function f(z) is defined as Re(f(z)) and is denoted by u(x, y), where z = x + iy.

Constant Real Part:
Since the real part of f(z) is constant, it means that Re(f(z)) = c, where c is a constant.

Implication:
If the real part of a function is constant, it implies that the function is constant.

Proof:
To prove that f(z) is constant, we can consider the Cauchy-Riemann equations. The Cauchy-Riemann equations are a necessary and sufficient condition for a complex function to be analytic.

The Cauchy-Riemann equations are given by:

∂u/∂x = ∂v/∂y ...(1)
∂u/∂y = -∂v/∂x ...(2)

where u(x, y) is the real part of f(z) and v(x, y) is the imaginary part of f(z).

Since the real part of f(z) is constant, the partial derivatives of u(x, y) with respect to x and y are both zero. This implies that the partial derivative of v(x, y) with respect to x and y are also zero.

Therefore, the Cauchy-Riemann equations are satisfied for all points in the domain of f(z), which means that f(z) is analytic.

Since f(z) is analytic and its real part is constant, it implies that f(z) is constant.

Conclusion:
Hence, the correct answer is option D: f(z) is constant.