on $D.$

Recall that a set $V\subseteq X$ is of first category in the sense of Baire, if $V=\bigcup_{n=1}^{\infty }V_{n}$, where $V_{n}\subseteq X,\,n\in{{\mathbb N}}$ are nowhere dense sets i. e. ${\operatorname*{int}}\left( {\operatorname*{cl}}(V_{n})\right) =\emptyset $.

Note that condition $$\forall\,x,y\in D\setminus C \mbox{ the set } [x,y]\cap C \mbox{ is countable, possibly empty}$$ imposed on the set $C$ in Proposition \[pp\], has been weakened in [@KaPiLa]. More precisely, in [@KaPiLa] it was established the following result.

\[ppp\]*(Theorem 3, [@KaPiLa])* Let $X$ be a real Banach space and let $V\subseteq X$ be a set of first category with the following property $$\forall x,y\in D\setminus V,\mbox{ the set }[x,y]\cap V\mbox{ is countable, possibly empty.}$$ Assume that $T:D{\longrightarrow}X^*$ is hemicontinuous and locally monotone on $D\setminus V$. Then $T$ is monotone on $D$.

Note that the conclusion of Proposition \[ppp\] remains valid if the condition $ \forall x,y\in D\setminus V,\mbox{ the set }[x,y]\cap V\mbox{ is countable, possibly empty}$ is replaced by the condition $$\forall x,y\in D\setminus V,\mbox{ the set }[x,y]\cap V\mbox{ contains no nonempty perfect subsets.}$$

\[r22\] It is worth mentioning that Proposition \[ppp\] cannot be extended to the set-valued case since, according to a theorem of Kenderov (see [@Ke], Proposition 2.6), a monotone operator that 