A Riemannian metric which doesn't vary smoothly A Riemannian metric $ds^2$ on a differential manifold $M$ of dimension $n\ge 2$ is written as $ds^2=\sum_{i=1}^n\sum_{j=1}^ng_{ij}(x_1,\dots,x_n)dx_idx_j$ where the functions $g_{ij}:M\rightarrow \mathbb{R}$ are twice continuously differentiable functions. 
The derivatives of the $g_{ij}$ are fundamental to define tools as Christoffel symbols, connection, curvature..
My question is: what happens if the $g_{ij}$ still admit derivatives of first and second order in every point of $M$, but these derivatives are not continuous? Is it still possible to define and use Christoffel symbols and curvature tensor (and other tool involving derivatives of $g_{ij}$) or should one expect something going wrong?
 A: "Smooth" is a tricky word in Mathematics.  Its meaning is very subject specific.  It is likely that your source at one point defines a Riemannian metric to be a collection of smooth functions and at another point defines "smooth" to be twice continuously differentiable.  Twice continuously differentiable is sufficient (since you do not use any properties of higher derivatives of the functions) for all the standard results.
Generalized Riemannian spaces relax the smoothness (and as a result, many standard results are weakened or altered).  Looking at my bookshelf, a random relevant text would be  Berestovskij, V.N. and I.G. Nikolaev, "Multidimensional Generalized Riemannian Spaces", appearing in Geometry IV: Non-regular Riemannian Geometry, which is Vol. 70 in the Encyclopaedia of Mathematical Science, 1993 (ISBN 13:  978-3642081255).  In this work, one looks at bounding the curvature of the space as a way to control wrongness.  The authors bound the curvature by starting with the first variation of the length in an intrinsic metric space.
A: Beyond the theory of spaces of curvature bounded above/below in the sense of Alexandrov, people also studied scalar curvature defined in the distributional sense. If you are interested in the 2-dimensional case, then the scalar curvature is all what you need: It becomes a (signed) measure on the surface. Take a look at, say, this paper to see how this is done:
Dan A. Lee, Philippe G. LeFloch, The positive mass theorem for manifolds with distributional curvature. 
In fact, they make even weaker regularity assumption than $C^1$: $g\in C^0\cap W_{loc}^{1,n}$ (where $M$ is $n$-dimensional). 
