how to find out which of the following holds? Let $f$ : $\mathbb R \to \mathbb R$ be a given function.
 Consider the following statements:
$A$: The function $f$ is continuous almost everywhere.
$B$: There exists a continuous function $g : \mathbb R \to \mathbb R$ such that $f = g$ almost everywhere.
Which of the following implications are true?
a) $A\Rightarrow B$
b)  $B\Rightarrow A$
I have no idea how to look for this?

Both implications are false.
1) suppose f is a.e. continuous. Take for example: $$f (x) = \frac{1}{x^2}$$ Now suppose that we have $$g: \mathbb{R} \rightarrow  \mathbb{R} $$ and suppose $ g $ is continuous.  Then if $g=f$ a.e. we get a contradiction,  because $ g $ has to explode in zero and hence it cannot be continuous. 
2) take $ f  = 1_{\mathbb{Q}}$ then $f=g$ a.e. but $ f $ has no points of continuity.
