Let
C
C
be a family of continuous commuting functions of an interval
I
I
into itself. If each function, except for possibly one, has an interval
[
a
,
b
]
,
a
⩽
b
[a,b],a \leqslant b
, for its set of fixed points or does not have periodic points except fixed ones, then it is shown that
C
C
has a common fixed point. This result generalizes a previous theorem of T. Mitchell.