Let \(C_0(\mathbb{R}^n)\) be the space of continuous functions vanishing at infinity . Then \(C_0(\mathbb{R}^n)\nsubseteq L^1(\mathbb{R}^n)\), since \(𝟙_{[-1,1]}+ \frac{1}{x}𝟙_{[-1,1]^c}\) is continuous non integrable function vanishing at infinity.
Let \(C_0(\mathbb{R}^n)\) be the space of continuous functions vanishing at infinity . Then \(C_0(\mathbb{R}^n)\nsubseteq L^1(\mathbb{R}^n)\), since \(𝟙_{[-1,1]}+ \frac{1}{x}𝟙_{[-1,1]^c}\) is continuous non integrable function vanishing at infinity.