Final answer:
The statement is true: if the real part of an entire function is bounded, then the function is constant, as per the maximum modulus principle which states that an entire non-constant function must have an unbounded modulus.
Step-by-step explanation:
The question asks whether it is true or false that if the real part of an entire function is bounded, then the function is constant. This statement is indeed true and can be shown using the maximum modulus principle. If an entire function, which is a complex function that is holomorphic everywhere on the complex plane, has a bounded real part, then by applying the principle, the entire function must be constant.
According to the maximum modulus principle, if a function is entire and non-constant, its modulus |f(z)| attains no maximum value on the complex plane. However, when the real part of f is bounded, we can conclude that |f(z)| is bounded as well because the modulus of a complex number is influenced by the magnitude of its real part. Therefore, if the real part of f does not exceed a certain value, neither will the modulus |f(z)|. But if |f(z)| is bounded in the entire complex plane, that contradicts the property of a non-constant entire function of having an unbounded modulus. Hence, we deduce that the function f must indeed be constant.