Question

\[f:[0,1] \rightarrow [0,1]\] is continous. Show that there exist a \[a \in [0,1]\] \[f(a)=a\]. (make use of intermediate theorem)

Answer 1

define the thrm ....

Answer 2

ok just a moment

Answer 3

\[f:[a,b]\rightarrow \mathbb{R}\] is continious, \[f(a)\] not equal to \[f(b)\] and there exist a value \[y_{0}\] between \[f(a)\] and \[f(b)\] . Then there exist \[x_{0} \in [a, b] \] with \[f(x_{0})=y_{0}\]

Answer 4

f(a) not equal to f(b) ?

Answer 5

yes

Answer 6

well, lets do some trial runs

f(x) = x = x^2 when x=1, or 0


can we find a point in here?