# Help in Limits of Continuity...

• Jun 17th 2008, 08:42 AM
Vedicmaths
Help in Limits of Continuity...
I need help in proving those problems.

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!
• Jun 17th 2008, 09:00 AM
TheEmptySet
Quote:

Originally Posted by Vedicmaths
I need help in proving those problems.

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!

2) define $g(x)=2^x-3x$ So g is continous on [0,1] and
$g(0)=1 \\\ g(1)=-1$ so by the intermediate value theorem There exisits a $c \in (0,1)$ such that $g(c)=0$

$g(c)=0 \iff 0=2^c-3c \iff 3c=2^c$

3) $f(x)=\frac{(x-5)(x+1)}{(x-5)} = x+1 \mbox{ if } x \ne 5$

so if we define $f(5)=6$ We get what we want :D
• Jun 17th 2008, 09:04 AM
Plato
For #1, note that between any two numbers there is an irrational number. Being continuous the function has the intermediate value property.
• Jun 17th 2008, 09:09 AM
red_dog
Suppose f is not constant. Then $\exists x_1,x_2\in[a,b], \ x_1\neq x_2$ such as $f(x_1)=\alpha, \ f(x_2)=\beta, \ \alpha\neq\beta$.
Let $c\in(\alpha, \beta)-\mathbf{Q}$.
f continuous $\Rightarrow \exists x_3\in(x_1,x_2)$ such as $f(x_3)=c$.
But, $f(x)\in\mathbf{Q} \ \forall x\in[a,b]$, contradiction.