The class of linearly continuous functions f:**R**^{n}-->**R**, that is,
having continuous restrictions f|L
to every straight line L, have been studied since the dawn of the twentieth century.
In this paper we refine a description of
the form that the sets D(f) of points of discontinuities of such functions can have.
It has been proved by Slobodnik that D(f) must
be a countable union of isometric copies of the graphs of Lipschitz functions
h:K-->**R**, where K is a compact nowhere dense subset of **R**^{n-1}.
Since the class **D**^{n} of all sets D(f), with f:**R**^{n}-->**R**
being linearly continuous,
is evidently closed under countable unions as well as under isometric images,
the structure of
**D**^{n} will be fully discerned upon deciding precisely which graphs of the
Lipschitz functions
h:K-->**R**,
K being compact nowhere dense subset of **R**^{n-1}, belong to **D**^{n}.
Towards this goal, we prove that **D**^{n} contains the graph of any such
h:K-->**R**
whenever h is a restriction of
convex function from **R**^{n-1}
into **R**. Moreover, for n=2, **D**^{2}
contains the graph of any such h, if h can be extended
to a
**C**^{2}
function H:**R**-->**R**.
At the same time, we provide an example, showing that this last result need not hold when H is just differentiable
with bounded derivative (so Lipschitz).

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**Last modified September 17, 2013.**