Analyzing the given options:
To understand why option B is the correct answer, let's analyze the given options:

a) Function of z: This option suggests that the function f(z) is a function of z, which means it depends on the complex variable z itself. However, the given information states that the modulus of f(z) is constant, which means that the function does not depend on the specific values of z but only on their magnitudes. Therefore, this option is not correct.

b) Constant: This option suggests that the function f(z) is a constant function, meaning it does not depend on z at all. Since the modulus of f(z) is constant, it implies that the function has a constant magnitude and does not vary with z. Therefore, this option is correct.

c) Function whose only imaginary part is constant: This option suggests that the function f(z) has a constant imaginary part but may have a non-constant real part. However, the given information states that the modulus of f(z) is constant, which means that both the real and imaginary parts must be constant. Therefore, this option is not correct.

d) Function whose only real part is constant: This option suggests that the function f(z) has a constant real part but may have a non-constant imaginary part. However, the given information states that the modulus of f(z) is constant, which means that both the real and imaginary parts must be constant. Therefore, this option is not correct.

Understanding the concept:
In complex analysis, a function is said to be analytic if it is differentiable at every point within its domain. The Cauchy-Riemann equations provide necessary and sufficient conditions for a function to be analytic.

For a function f(z) to have a constant modulus, it means that |f(z)| = constant for all values of z in its domain. Geometrically, this represents a set of points on the complex plane that lie on a circle or a straight line.

Why option B is correct:
If the modulus of f(z) is constant, it implies that the function maps every point in its domain to a fixed distance from the origin on the complex plane. This means that the function does not change its magnitude as z varies.

Since the magnitude of a complex number is determined by both its real and imaginary parts, for the modulus to be constant, both the real and imaginary parts of f(z) must be constant. Therefore, the correct answer is option B: Constant.

Conclusion:
In summary, if f(z) is an analytic function whose modulus is constant, then the function must be a constant function, meaning it does not depend on the variable z. Both the real and imaginary parts of f(z) must be constant for the modulus to remain constant.