We suppose that $X$ is a locally convex vector space. Before showing that the conjecture is true (under some conditions), we show a helpful lemma.
Lemma. If a subset $C$ of $X$ has the property that it's boundary can be covered by an open cover $\{U_\alpha\}_{\alpha\in A}$ with the property that each $U_\alpha\cap C$ is convex, then the triangle
$$
\Delta(x,y,z):=\{(1-t)((1-s)x+sy)+tz:s,t\in[0,1)\}
$$
is contained in $C$ whenever the lines $\ell_{x,y}:=\{(1-s)x+sy:s\in[0,1]\}$ and $\ell_{x,z}:=\{(1-t)x+tz:t\in[0,1]\}$ are contained in $C$.
Proof. Suppose that $\Delta(x,y,z)$ is not contained in $C$. In the following, the boundary and interior are considered relative to the triangle $\Delta(x,y,z)$.
Among $s\in[0,1)$ with the property that the line
$$
\{(1-t)((1-s)x+sy)+tz:t\in[0,1)\}\cap \partial(C\cap\Delta(x,y,z))\neq\varnothing
$$
there is a minimum $s_0$. Then there is also a minimum $t_0$ among the $t\in[0,1)$ with the property that
$$
u:=(1-t)((1-s_0)x+s_0y)+tz:t\in\partial(C\cap\Delta(x,y,z))
$$
Since $u$ is on the boundary of $C$, there exists an $\alpha\in A$ with the property that $u\in U_\alpha$. Then $U_\alpha\cap C\cap\Delta(x,y,z)$ is convex. But note that the set
$$
\ell_{x,y}\cup\ell_{x,z}\cup\Delta(x,(1-s_0)x+s_0y,z),
$$
where the latter triangle has a similar definition as $\Delta(x,y,z)$, is contained in $C$. The convexity of $U_\alpha\cap C\cap\Delta(x,y,z)$ then implies that there is a $\varepsilon>0$ such that the point
$$
(1-(t_0-\varepsilon))((1-s_0)x+s_0y)+(t_0-\varepsilon)z
$$
is on the boundary of $C\cap\Delta(x,y,z)$ unless $t_0=0$. The minimality of $t_0$ then gives us that $t_0=0$. A similar argument shows that $s_0=0$ and it follows once again from the convexity of the boundary of $C$ that $x,y,z$ are on a line. But then $\Delta(x,y,z)$ is (contained in) this line. QED.
We have the following corollary (in the open case, an easy adjustment of the above proof has to be given).
Corollary. If $X$ is either open or closed, the triangle
$$
\overline{\Delta(x,y,z)}=\{(1-t)((1-s)x+sy)+tz:s,t\in[0,1]\}
$$
is contained in $C$.
Now we are ready to answer positive to the conjecture in the case that $C$ is open or closed.
Proposition. Suppose that $X$ is a locally convex vector space and suppose that $C$ is a closed [open] subset of $X$. If there exists a cover $\{U_\alpha\}_{\alpha\in A}$ of $\partial C$ with the property that $U_\alpha\cap C$ is convex for each $\alpha\in A$, then $C$ is convex.
Proof.
Suppose $C$ is a subset of a locally convex vector space $X$ and that $\{U_\alpha\}_{\alpha\in A}$ is a cover of the boundary of $C$ with the property that $U_\alpha\cap C$ is convex for each $\alpha\in A$.
First, let $B$ be a maximal among the subsets of $A$ with the property that the convex hull of $\bigcup_{\alpha\in B}U_\alpha$ is contained in $C$. For example, such a maximal $B$ can be found with Zorn's lemma, using the poset
$$
P:=\big\{B\subseteq A:\mathrm{Hull}\big(\textstyle\bigcup_{\alpha\in B}U_\alpha\cap C\big)\subseteq C\big\}
$$
ordered with inclusion. Here, $\mathrm{Hull}$ denotes the operation of taking the convex hull.
Now let $C^\prime$ be a maximal convex subset of $C$ which contains $U_\alpha$ for each $\alpha\in B$. Here's an outline of the rest of the proof.
- First we will show that if for $\alpha\in A$ there is a $\beta\in B$ such that $U_\alpha\cap U_\beta\cap C\neq\varnothing$, then $\alpha$ is in $B$.
- Then we will show that no boundary points of $C^\prime$ are interior points of $C$. (Here we will use the assumption that $X$ is locally convex.)
With these two observations the proof is easily finished. Since boundary points of $C^\prime$ are boundary points of $C$, it follows that $C^\prime$ is closed in $C$. The set
$$
U:=\textstyle\bigcup_{\alpha\in B} U_\alpha\cup\mathrm{int}\,C^\prime
$$
is an open set with the property that $x\in C^\prime$ whenever $x\in U\cap C$, therefore $C^\prime$ is also open in $C$. The connectedness of $C$ together with the non-emptiness of $C^\prime$ (whenever $C$ is non-empty) now implies that $C^\prime= C$.
Proof of 1. Suppose that $x\in U_\alpha\cap U_\beta\cap C$. Note that for any $y\in U_\alpha$, for any $\gamma\in B$ and for any $z\in U_\gamma$, the line
$$
\ell_{y,z}:=\{(1-t)y+tz:t\in[0,1]\}
$$
is contained in $C$, which is a consequence of the corollary because the we have $\ell_{x,y}\subseteq U_\alpha\cap C\subseteq C$ and $\ell_{x,z}\subseteq C^\prime\subseteq C$. By the maximality of $B$, it follows that $\alpha\in B$.
Proof of 2. Suppose that $x^\prime$ is in $\partial C^\prime\cap\mathrm{int}\,C$. Let $V$ be a convex open neighborhood of $x^\prime$ which is contained in $C$, let $y$ be a point of $V\setminus \bar{C^\prime}$ and let $x$ be a point of $V\cap C^\prime$. It follows by the corollary that for every $z\in C^\prime$ the line $\ell_{y,z}$ is contained in $C$, since the lines $\ell_{x,y}$ and $\ell_{x,z}$ are. Therefore, $C^\prime$ can be extended to include $y$, which contradicts the maximality of $C^\prime$. Thus we may conclude that no boundary points of $C^\prime$ are interior points of $C$. QED.
Best Answer
No. Just add any continuous, strictly convex function (e.g. the norm squared) to a discontinuous linear functional. The sum of a convex $f$ and strictly convex $g$ must be strictly convex, as \begin{align*} (f + g)(\lambda x + (1 - \lambda)y) &= f(\lambda x + (1 - \lambda)y) + g(\lambda x + (1 - \lambda)y) \\ &\le \lambda f(x) + (1 - \lambda)f(y) + g(\lambda x + (1 - \lambda)y) \\ &< \lambda f(x) + (1 - \lambda)f(y) + \lambda g(x) + (1 - \lambda)g(y) \\ &= \lambda (f + g)(x) + (1 - \lambda)(f + g)(y) \end{align*} for distinct $x, y \in E$ and $\lambda \in (0, 1)$. Also, if $f + g$ were continuous at any point, then so would be $f = (f + g) - g$, by the algebra of continuous functions.