- Warm-up-exercises
Show that a
linear function
-
is continuous.
Prove that the function
-
is continuous.
Prove that the function
-
is continuous.
Let
be a subset and let
-
be a continuous function. Let
be a point such that
.
Prove that
for all in a non-empty open interval .
Let
be real numbers and let
-
and
-
be continuous functions such that
.
Prove that the function
-
such that
-
is also continuous.
Compute the limit of the sequence
-
for .
Let
-
be a continuous function which takes only finitely many values. Prove that is constant.
Give an example of a
continuous function
-
which takes exactly two values.
Prove that the function
-
defined by
-
is only at the zero point continuous.
Let
be a subset and let
be a point. Let
be a function and
.
Prove that the following statements are equivalent.
- We have
-
- For all
there exists a
such that for all
with
the inequality
holds.
- Hand-in-exercises
We consider the function
-
Determine the points
where is
continuous.
Compute the limit of the sequence
-
where
-
Prove that the function
defined by
-
is for no point
continuous.
Decide whether the sequence
-
converges and in case determine the limit.
Determine the
limit
of the
rational function
-
in the point
.