From my understanding, the gradient vector field on a Riemannian manifold $M$ is usually defined for a smooth function $f \colon M \to \mathbb{R}$ (e.g., John Lee's Introduction to Riemannian Manifolds), while it seems natural to define a gradient vector field of a $C^1$ function defined in $\mathbb{R}^n$.
Does defining the gradient vector field of a $C^r$ ($r \neq \infty$) function (e.g., $C^1$ function) on a Riemannian manifold cause any inconvenience? If so, what is the inconvenience? If not, is there any textbook where the gradient for a not $C^{\infty}$ function is defined?