Uniform continuity of $h$ satisfying $ h(a,y) \geq \alpha$ for al $y$ and $a$ is fixed

Question

Let $h:\mathbb{R}^d \times \mathbb{R}^n \to \mathbb{R}$ be a $C^1$ mapping and $a\in \mathbb{R}^d$. Assume that there is $\alpha>0$ such that $$ h(a,y) \geq \alpha,$$ for all $y\in \mathbb{R}^n$.

Can I deduce by continuity of $h$ that there is a neighborhood of $a$ such that $$ h(x,y) \geq \frac{1}{2}\alpha,$$ for all $(x,y)\in \overline{B}(a,\beta)\times\mathbb{R}^n$ for some $\beta>0$.

Or should I have $y$ belong to a compact set?