Function that satisfies Polyak Lojiasiewicz inequality but not coercive

Question

I am wondering if it is possible to have a function that is not coercive, bounded below and satisfies the famous Polyak Lojiasiewicz inequality $\lVert \nabla f(x)\rVert^2 \geq 2\mu(f(x)-f^*)$ for all $x\in\mathbb{R}^n$. In other words, I am not sure whether Polyak Lojiasiewicz inequality would actually imply coerciveness although it does not imply convexity.

Answer 1

Every constant function is an example.

As another example, consider $f(x)=\left(\max\{x,0\}\right)^2$ which is not coercive. Then $f^*=\min_{x\in\mathbb{R}}f(x)=0$ and $f'(x)=\max\{2x,0\}$. In this case, the Polyak Lojiasiewicz inequality holds with $\mu=2$: $$ \left(f'(x)\right)^2=4\left(\max\{x,0\}\right)^2=4(f(x)-f^*). $$