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A detailed solution is then proposed to you.
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Question
We denote by
an interval of
and by
and
two applications defined from
to
that are continuous in a point
a) Prove that application
is continuous in
b) Prove that if application
is continuous in a point
such that application
is bounded on
c) Prove that application
is continuous in
a) Write the definition of the continuity of both applications in
with
then use the triangular inequality.
b) Write the definition of the continuity in
with
then use the triangular inequality with the form
c) Start from:
then use the triangular inequality then the previous question.
a) Let
be a strictly positive real number.
Functions
and
being continuous in
there exist two strictly positive real numbers
and
such that:
We then set
and using the triangular inequality, we obtain:
This reasonning can be made for any strictly positive real number
hence we proved that function
is continuous in
b) We set
and write the definition of continuity in
being an open interval and
we can choose
small enough such that
in included in
We then use the triangular inequality :
which implies:
meaning:
therefore application
is bounded on
c) With the triangular inequality, we obtain:
We consider an interval
included in
Application
is bounded on interval
hence there exists a real number
which we can assume is non zero, such that:
Similarly, there exists a real non zero number
such that:
We then set
Let
be a strictly positive real number.
Functions
and
being continuous in
there exists two real numbers
and
such that:
We also have:
We then set
and we obtain:
.
The strictly positive real number
being arbitrary, this infers that function
is continuous in
