Thread: Help in Limits of Continuity...

I need help in proving those problems.
Please help..

Q1) Suppose f: [a,b] ->R is continuous and that f([a,b]) is subset of Q. Prove that f is constant on [a,b] using Intermediate value Theorem.

Q2) Prove that 2^x = 3x for some x E (o,1).

Q3) Let f(x) = (x^2 - 4x - 5) / (x-5) for x not equal to 5. Can f(5) be defined so as to make f continuous at 5.


Thanks!

Originally Posted by Vedicmaths

I need help in proving those problems.
Please help..

Q1) Suppose f: [a,b] ->R is continuous and that f([a,b]) is subset of Q. Prove that f is constant on [a,b] using Intermediate value Theorem.

Q2) Prove that 2^x = 3x for some x E (o,1).

Q3) Let f(x) = (x^2 - 4x - 5) / (x-5) for x not equal to 5. Can f(5) be defined so as to make f continuous at 5.


Thanks!

Still thinking about 1)

2) define So g is continous on [0,1] and
so by the intermediate value theorem There exisits a such that



3)

so if we define We get what we want

For #1, note that between any two numbers there is an irrational number. Being continuous the function has the intermediate value property.

Suppose f is not constant. Then such as .
Let .
f continuous such as .
But, , contradiction.