Differentiability and Derivatives

The function $f$ is Gateaux differentiable at $x$ if and only if the Dini directional derivative $\delta f(x,z)$ exists for all $z$ and is linear in $z$. Thus $\delta f(x,z)=G(x)z$

The function $f$ is Hadamard differentiable at $x$ if the Hadamard directional derivative $df(x,z)$ exists for all $z$ and is linear in $z$.

Function $f$ is locally Lipschitz at $z$ if there is a ball $\mathcal{B}(z,\delta)$ and a $\gamma>0$ such that $\|f(x)-f(y)|\leq\gamma\|x-y\|$ for all $x,y\in\mathcal{B}(z,\delta).$

If $f$ is locally Lipschitz and Gateaux differentiable then it is Hadamard differentiable.

If the Gateaux derivative of $f$ is continuous then $f$ is Frechet differentiable.

Define Frechet differentiable

The function $f$ is Hadamard differentiable if and only if it is Frechet differentiable.

Gradient, Jacobian