Recall that fact that a distribution which is compactly supported can be written as a finite sum of derivatives of continuous functions. This fails for general distributions in $D^\prime (\mathbb{R})$. My idea for a counterexample was $$u:=\sum_{n\in\mathbb{N}}\delta_n,$$ where $\delta_n(\phi) = \phi(n)$ for a test function $\phi\in D(\mathbb{R})= C_c^\infty(\mathbb{R})$.
Clearly, we have $\delta_n = H(\cdot - n)^\prime = | \cdot - n|^{\prime\prime}$, and thus $$u=\sum_{n\in\mathbb{N}} |\cdot -n|^{\prime\prime},$$ where $H$ denotes the Heaviside function. So it can be written as a countable sum of continuous functions.
However, how to show that it cannot be written as a finite sum of continuous functions and is this example actually correct?