# 2nd order approximation for nondifferentiable convex function

Consider $$f$$ to be a convex multivariate function from $$\mathbb{R}^N$$ to $$\mathbb{R}$$, not differentiable (but, by definition, left- and right-differentiable).

Does some kind of "directional" 2nd-order approximation formula exist, for example where $$f(x+ty) = f(x)++\frac{1}{2}$$

for $$(x,y)\in\mathbb{R}^N$$ and a scalar $$t>0$$ and $$v$$ is a subgradient of $$f$$ at $$x$$ and $$A$$ is a matrix? When $$f$$ is convex not differentiable, I struggle to justify that.

Thanks for your help.