I'm studying the book by Ambrosio on gradient flows in metric spaces. It's stated that the usual notion of gradient flow, $$ x'(t) = -\nabla_x f(x),$$ is not defined on metric spaces because we don't have a definition for $x'(t)$ nor $\nabla_x$. This is stated as obvious but it's not 100% clear to me.
I guess that since derivatives on a metric space would be defined as $$ f'(x) = \lim_{h\downarrow 0} \frac{d(f(x), f(x+h))}{h}$$ (where $d$ is a metric), we need to ensure that limits exist in the metric space. I thought of the discrete metric space where $d(x,y) = 0$ if $x=y$ and $1$ if $x\neq y$. But in this case, we can't define differentiable functions other than ones that are constant on singleton points. Is there a better example?
Another thing that might be needed to define derivatives is the idea of direction. Since the gradient is essentially a set of directional derivatives in the basis vectors of the underlying space (correct me if this is wrong?), we may not be able to define this in a metric space that doesn't have "directions".
What are some pathological examples of metric spaces that don't have directions or for other reasons we can't define derivatives? Is there anything else preventing derivatives/gradients from being defined in metric spaces?