Suppose $\lambda \in C_c^\infty(\mathbb{R})^*$ is a distribution and $f: \mathbb{R} \to \mathbb{R}$ is a continuous map of the real line. In addition assume $f$ has compact support. How can I make sense of $f\circ \lambda$? I would like to define it by $\langle \phi, f\ \circ \lambda \rangle := \langle \phi \circ f, \lambda\rangle \text{, for } \phi \in C^\infty_c(\mathbb{R}),$ but $\phi \circ f$ is only continuous becasue $f$ is only assumed to be continuous.
My question is:
Does it makes sense to define $\langle \phi \circ f, \lambda\rangle$ through approximation of $f$ by smooth functions $f_i$? That is: $\langle \phi \circ f, \lambda\rangle =\lim_{i \rightarrow \infty} \langle \phi \circ f_i, \lambda\rangle ?$