Let $f: [0, 1] \to \mathbb R$ be a measurable function. A function $g: [0, 1] \to \mathbb R$ is said to be a condensation limit of $f$ if $g$ is continuous and agrees with $f$ on a dense subset of $[0, 1]$.

Let $k \geq 1$ be an integer, and $f: [0, 1] \to \mathbb R$ be a measurable function whose graph is dense in $[0, 1] \times \mathbb R$.

Question: Is the set of $k$ times continuously differentiable condensation limits of $f$ dense in $C^k$?

Note: Here $C^k$ denotes the space of $k$ times continuously differentiable functions under the norm $\| g \|_{C^k} := \sum_{i = 0}^k \sup _{x \in [0, 1]} \lvert g^{(i)} (x) \rvert$, where $g^{(i)}$ is the $i$’th derivative of $g$.