 Solution

For a random sample, write $g_j(x)={\prod }_{i=1}^{n}f(x_i;{\theta }_j)$, $j=0,1$, for its joint probability density functions. Let $\phi$ be any competing statistical test of level at most $\alpha$. Pointwise,Indeed, where $g_1>k{g}_0$, ${\phi }_{\ast }=1\ge \phi$; where $g_1<k{g}_0$, ${\phi }_{\ast }=0\le \phi$, so both factors have the same sign. On the equality set the product is zero, regardless of randomization. Integrate over the joint sample space using its ordinary volume measure:Since $k\ge 0$ and ${\mathbb{E}}_0{\phi }_{\ast }=\alpha \ge {\mathbb{E}}_0\phi$,This proves maximal power. A continuous sample distribution does not itself guarantee that the equality set of the likelihood ratio has zero probability; allowing randomization there makes the proof valid in that case too.