This is just a prework.
Given a compact domain.
Regard the function space: $$\mathcal{C}(\Omega,\mathbb{R}):=\{f:\Omega\to\mathbb{R}:f\text{ continuous}\}$$
Clearly it is an algebra: $$f+g\in\mathcal{C}(\Omega,\mathbb{R})\quad\lambda f\in\mathcal{C}(\Omega,\mathbb{R})$$
How to see that it is a lattice?
This answer is community wiki.
The meet and join can be rewritten as: $$f\wedge g=\frac12\{(f+g)-|f-g|\}\quad f\vee g=\frac12\{(f+g)+|f-g|\}$$ (Note that addition, multiplication and modulus are continuous.)