Let $M$ be a Riemannian manifold; if $d$ is the distance on $M$, we can consider the distance $D$ between any two continuous curves given by $D(c, \gamma) = \max \_{t \in [0,1]} d(c(t), \gamma(t))$.



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> Let $c:[0,1] \to M$ a continuous curve. Is it true that for every $\varepsilon > 0$ there exists a differentiable (or even smooth) curve $\gamma$ such that $D(c, \gamma) < \varepsilon$?
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I believe that this is true, and I also believe that I have proven it (the case of Peano-like curves make me doubt my reasoning, though). The idea is to consider sufficiently many points on $c$, join each two consecutive of them by a unique minimizing geodesic, join all these geodesic segments in a single zig-zag line and show that this line is at distance at most $\varepsilon$ from $c$. The problem is that its proof is 1.5 pages long (a lot of playing around with the injectivity radius), technical and boring. It would help me if, instead of including it in an article that I am writing, I could replace it with just a reference.



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> Assuming the result above is true, can you please provide a citable reference?
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There exists differentiable curves arbitrarily close to the continuous ones
It turns out that something much more general is true and can be found in the literature.



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> **Theorem [Thm 3.3, Hirsch, *Differential Topology*]** Let $M$ and $N$ be $C^s$-manifolds (with boundary), $1\le s\le\infty$. Then $C^s(M,N)$ is dense in $C\_S^r(M,N)$ for $0\le r<s$, where $C\_S^r(M,N)$ is equipped with the "strong" topology (when $r=0$, it becomes the compact-open topology).
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In particular, when $s=\infty$, $r=0$, $M$ is compact and $N$ is equipped with a (Riemannian) metric, it leads to the result that you want.


