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Show that f is proper if and only if f* is continuous

Let X and Y be locally compact Hausdorff spaces.
Let X* and Y* be their one point compactifications.
Let f be a continuous map from X to Y. Let f* be the map from X* to Y* whose restriction to X is f, and which takes the point at infinity in X* to the point at infinity in Y*.
Show that f is proper if and only if f* is continuous.

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The attached file is half a page written in Word with full mathematical notation, comprising a full step by step solution to the question. Included are the appropriate definitions and elementary results required to answer the question.

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