Continuity function bounded on half-line

I need to prove the following theorem:

Let $f: \mathbb{R} \to Y$, where $Y$ is a normed space, be an additive function. Suppose there exists an interval $(a, +\infty) \subset \mathbb{R}$ such that there exists a constant $M > 0$ satisfying:$$\|f(t)\| \geq M \quad \text{for every } t \in (a, +\infty).$$The goal is to prove that the function $f$ is either linear or continuous.

My Attempts: I have already shown that any additive function is $\mathbb{Q}$-homogeneous, i.e., for any $q \in \mathbb{Q}$,$$f(qt) = qf(t).$$

To proceed further, I attempted to define a new function:$$g(t) = f(t) + c \cdot t, \quad \text{where } c \in \mathbb{R}.$$This function satisfies the following properties:

1. $g(t)$ is odd, i.e., $g(-t) = -g(t)$,
2. $g(t)$ is homogeneous, i.e., for any $a \in \mathbb{Q}$,$$g(at) = ag(t).$$

Using the given bound $\|f(t)\| \geq M$, I need to show that:$$g(t) = 0.$$

If I can prove this, then it follows that $f$ is homogeneous over $\mathbb{R}$. Combining this with additivity, it will imply that $f$ is continuous.

Request for Help: Unfortunately, I am struggling with proving that $g(t) = 0$. I would greatly appreciate any hints or a complete solution to this problem.