Is a linear operator that is tangent to a continuous function from a TVS to a Riesz space also continuous?

Question

Let $\mathcal{X}$ be a topological vector space and $\mathcal{Y}$ be a topological Riesz space. Suppose $f:\mathcal{X} \rightarrow \mathcal{Y}$ is continuous at $0_{\mathcal{X}}$ and has $f(0_{\mathcal{X}}) = 0_{\mathcal{Y}}$. Suppose $a:\mathcal{X} \rightarrow \mathcal{Y}$ is a linear operator and $a \leq f$ point-wise. Is $a$ guaranteed to be continuous?

Are more assumptions needed, e.g. that $\mathcal{Y}$ is locally bounded?