Let f(x)=[x] and . Then
g(x) exists, but g(x) is not continuous at x=2.
f(x) does not exist and f(x) is not continuous at x=1.
gof is continuous for all f(x)
fog is continuous for all x.
Since, and g(1)=0. So
g(x) is not continuous at x=1 but g(x) exists
We have,
and,
So, does not exist and so f(x) is not continuous at x=1.
We have,
So, gof is continuous for all x.
We have,
which is clearly not continuous.
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