Needing help with one proof I've started self-studying convex analysis from Rockafellar's book recently. Upon proceeding to the Chapters 23-25, I've encountered some problems and could use some help: the author makes some statements while proving two theorems without offering a detailed account of why is that so:
Theorem 23.1.
Let $f$ be a convex function, and let $x$ be a point in where $f$ is finite. For each $y$, the difference quotient in the definition of $f'(x; y)$ is a non-decreasing function of $\lambda >0$ (WHY? IS IT NOT RATHER NON-INCREASING?), so that $f'(x; y)$ exists and
\begin{equation*}
f'(x; y) = \inf_{\lambda > 0} \frac{f(x+\lambda y) - f(x)}{\lambda}.
\end{equation*}
Moreover, $f'(x; y)$ is a positively homogeneous function of $y$, with $f'(x;0)=0$ and
\begin{equation*}
-f'(x;-y) \leq f'(x;y), \quad \forall y.
\end{equation*}
Proof.
The difference quotient for $\lambda >0$ can be expressed as $\lambda^{-1} h (\lambda y)$, where $h(y) = f(x+y) - f(x)$. The convex set $\text{epi}h$ is obtained by translating $\text{epi}f$ so that the point $(x,f(x))$ is moved to $(0,0)$. On the other hand $\lambda^{-1} h (\lambda y)=(h \lambda^{-1})(y)$, where by definition $h \lambda^{-1}$ is the convex function whose epigraph is $\lambda^{-1}\text{epi}h$. Since $\text{epi}h$ contains the origin, the latter set increases, if anything, as $\lambda^{-1}$ increases. (WHY?) ...

With respect to the last "why?":
Let $\lambda^{-1}<\delta^{-1}$ and let $(x,\mu) \in \lambda^{-1}\text{epi h}, i.e. (\lambda x,\lambda \mu) \in \text{epi h}$.
Note that $\kappa:=\frac{\lambda^{-1}}{\delta^{-1}}<1$. As $\text{epi h}$ is convex and contains the origin $(0,0)$ and the point $(\lambda x,\lambda \mu)$ it also contains the point 
$$(1-\kappa)(0,0)+\kappa(\lambda x,\lambda \mu)=\delta (x,\mu)$$
But the last statement is equivalent to $(x,\mu) \in \delta^{-1}\text{epi h}$
