  We study various boundary and inner regularity questions for
$p(\cdot)$-(super)harmonic functions in Euclidean domains. In particular, we
prove the Kellogg property and introduce a classification of boundary points
for $p(\cdot)$-harmonic functions into three disjoint classes: regular,
semiregular and strongly irregular points. Regular and especially semiregular
points are characterized in many ways. The discussion is illustrated by
examples. Along the way, we present a removability result for bounded
$p(\cdot)$-harmonic functions and give some new characterizations of $W^{1,
p(\cdot)}_0$ spaces. We also show that $p(\cdot)$-superharmonic functions are
lower semicontinuously regularized, and characterize them in terms of lower
semicontinuously regularized supersolutions.
