Why do I need compactness of domain and continuity of $Q$ to show the following?

Question

We are asked to show that

1. $B$ is the domain of $Q$ and is compact
2. $Q(\beta)$ is continuous in $\beta$
3. $Q(\beta)$ is uniquely minimized at $\beta_0$

implies

$\forall \epsilon >0 \; \exists \delta>0 : \underset{||\beta - \beta_0||>\epsilon}{\inf} Q(\beta) \geq Q(\beta_0) + \delta$.

Why do I need conditions 1. and 2. ? Am I not immediately finished using condition 3.?

Answer 1

I think condition $3$ is just meant to say that if the minimizer $\beta_0$ exists, then it is unique.

The first two conditions can be used to show that the minimizer exists.