Chain rule

Theorem: Assume that $g \colon \mathbb{R} \rightarrow \mathbb{R}$ is continuous, $g \colon \mathbb{T} \rightarrow \mathbb{R}$ is $\Delta$-differentiable and $f \colon \mathbb{R} \rightarrow \mathbb{R}$ is continuously differentiable. Then there exists $c$ in the interval $[t,\sigma(t)]$ with $$(f \circ g)^{\Delta}(t) = f'(g(c))g^{\Delta}(t).$$

Proof: █

Theorem: Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be continuously differentiable and suppose $g \colon \mathbb{T} \rightarrow \mathbb{R}$ is $\Delta$-differentiable. Then $f \circ g \colon \mathbb{T} \rightarrow \mathbb{R}$ is $\Delta$-differentiable and the formula $$(f \circ g)^{\Delta}(t) = \left\{ \displaystyle\int_0^1 f'(g(t)+h\mu(t)g^{\Delta}(t)) dh \right\} g^{\Delta}(t)$$ holds.

Proof: █