$f(\theta)<f(\theta^*)$ $\forall \theta\in \Theta$ implies $\theta^*=argmax_{\theta\in \Theta}f(\theta)$

Question

Take $\theta^*\in \Theta\subseteq \mathbb{R}^K$ and let $f$ be a real function of $\theta$.

If $f(\theta)<f(\theta^*)$ $\forall \theta\in \Theta$, with $\theta\neq \theta^*$, then does the following hold?

$$\theta^* = \arg\max_{\theta\in \Theta}f(\theta) \tag{1}$$

If not, which other conditions are sufficient to claim (1)? E.g., $\Theta$ closed and compact?

Answer 1

I think that you need some hypothesis on $f$, else the problem cannot be solved. If $f$ is continuous $\Theta$ compact is sufficient (every compact set is closed).