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{\bf Question}

Show that if $f$ is continuous then $\{x|f(x)\leq c\}$ is a closed
set.  Deduce that $f$ is measurable.


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{\bf Answer}

Let $a$ be a point of accumulation of $A=\{x|f(x)\leq c\}$.  Then
there exists $a_i\epsilon A, \,\,\,\, a_i\to a, \,\,\,\,
f(a_i)\leq c$ and $f$ is continuous, so $f(a_i)\rightarrow f(a)$
therefore $f(a)\leq c$.  i.e. $a\epsilon A$.

Therefore $A$ is closed and so measurable, thus $f$ is measurable.


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