Suppose a function of the standard sphere
S
2
{S^2}
into the standard sphere
S
2
+
m
{S^{2 + m}}
,
m
⩾
0
m \geqslant 0
, sends every circle into a circle but is not a circlepreserving bijection of
S
2
{S^2}
. Then the image of the function must lie in a five-point set or, if it contains more than five points, it must lie in a circle together with at most one other point. We prove the local version of this theorem together with a generalization to n dimensions. In the generalization, the significance of 5 is replaced by
2
n
+
1
2n + 1
. There is also proved a 3-dimensional result in which, compared to the n-dimensional theorem, we are allowed to weaken the structure assumed on the image set of the function.