- Warm-up-exercises
Find a zero for the function
-
in the interval
![{\displaystyle {}[0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5d0a058030cf6f54dea8f7b56f08f18cfa7e4f)
using the interval bisection method with a maximum error of

.
Let
-
be a continuous function. Prove that

is not surjective.
Give an example of a bounded interval
and a continuous function
-
such that the image of

is bounded, but the function admits no maximum.
Let
-
be a continuous function. Show that there exists a continuous extension
-
of
.
Determine directly, for which
the power function
-
has an extremum at the point zero.
Show that the Intermediate value theorem for continuous functions from
to
does not hold.
Determine the limit of the sequence
-
- Hand-in-exercises
Determine the minimum of the function
-
Find for the function
-
a zero in the interval
![{\displaystyle {}[0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5d0a058030cf6f54dea8f7b56f08f18cfa7e4f)
using the interval bisection method, with a maximum error of

.
Determine the limit of the sequence
-
The next task uses the notion of an even and an odd function.
A
function
-
is called
even, if for all

the equality
-
holds.
A function
-
is called
odd, if for all

the equality
-
holds.
Let
-
be a
continuous function.
Show that one can write
-
with a continuous
even function
and a continuous
odd function
.
The following task uses the notion of fixed point.
Let
be a set and let
-
be a function. An element

such that

is called a
fixed point of the function.
Let
-
be a continuous function from the interval
![{\displaystyle {}[a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e5a6e660b5cdc1a9c76f6e6b0d486328c3f6937)
into itself. Prove that

has a fixed point.