Neyman-Pearson $\alpha$-level test always exists for continuous distributions

Question

The Neyman-Pearson lemma as in the classical book by Casella and Berger, gives to conditions for the existence of $\alpha$-level tests:

1. The critical region must be of the form: $\{x:f(x|\theta_1) > k f(x|\theta_0)\} \subseteq R \subseteq \{x:f(x|\theta_1) \ge k f(x|\theta_0)\}$
2. $\alpha = P_{\theta_0}(X \in R)$

One can see that not every $\alpha$ has a suitable critical region such that 2. holds. The problem comes with discrete distributions.

Example

Let $X \sim Binom(2,\theta)$ and the test $H_0: \theta = 1/2$, $H_1:\theta = 3/4$. In this situation I cannot find a critical region of the form $\{x:f(x|\theta_1)/f(x|\theta_0) \ge k\}$ such that the size of the test is $\alpha = 1/\pi$ the reason is that all computations are done with rational values.

I had an argument that tried to proved that for continuous distributions the test of level $\alpha$ always exists but thanks to @Nch I found that it was wrong.

How can I prove that for continuous distributions there is always an $\alpha$-level test?