Let be any function. Define
.Then
is
g may be bounded even if f is unbounded
one-one if f is one
continuous if f is continuous
differentiable if f is differentiable
Take . Since
given by
is not one-one so (b) is violated. Also g is not differentiable at
. Let
then u is continuous function and
.Hence
is continuous if
is continuous. It is easy to see
is bounded if and only if
is bounded.
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