 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2009/e/1/b/ii/solution

Let

E = {x : f (x) \neq = f (a)}

be countable. For a closed subset

C

of the real line, if

f (a) \in C

, its inverse image contains

a

and is closed by definition. If

f (a) \in / C

, its inverse image lies inside

E

and is countable, hence is again closed. Every closed inverse image is therefore closed, proving $f$ is continuous.

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