## Characterizing continuity by preserving
compactness and connectedness

**János Gerlits, István Juhász, Lajos Soukup and
Zoltán Szentmiklóssy**

Let us call a function from a space into a space
*preserving* if the image of every compact subspace of
is compact in and the image of every connected subspace of
is connected in . By elementary theorems a continuous function
is always preserving. Evelyn R. McMillan proved in 1970
that if is Hausdorff, locally connected and Frèchet, is
Hausdorff, then the converse is also true: any preserving function
is continuous.
The main result of this paper is that if is any product of
connected linearly ordered spaces (e.g. if
) and
is a preserving function into a
regular space , then is continuous.

**Key words and phrases**:
Hausdorff space, continuity, compact, connected, locally
connected, Fr\`echet space, monotonically normal, linearly ordered space

**2000 Mathematics Subject Classification**:
54C05, 54D05, 54F05, 54B10

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