# right limit of a continuous bounded function

• Jun 8th 2011, 06:44 AM
abhishekkgp
right limit of a continuous bounded function
i was thinking whether the following is true or false:

Let $f$: $(a,b)$ $\rightarrow \mathbb{R}$ be a continuous and bounded function. then, $f$ has a right limit at $a$.(true or false?).

i thought that this could be proved to be true if i were able to show that the function is monotonic in some deleted right neighborhood of a but then $sin(1/x)$ served as a counter example as the claim would be violated at $x=0$.
intuitively it seems that the right limit at $a$ should exist but i am not able to prove it. please help.
• Jun 8th 2011, 07:11 AM
Plato
Quote:

Originally Posted by abhishekkgp
i was thinking whether the following is true or false: Let $f$: $(a,b)$ $\rightarrow \mathbb{R}$ be a continuous and bounded function. then, $f$ has a right limit at $a$.(true or false?).

i thought that this could be proved to be true if i were able to show that the function is monotonic in some deleted right neighborhood of a but then $sin(1/x)$ served as a counter example as the claim would be violated at $x=0$.
intuitively it seems that the right limit at $a$ should exist but i am not able to prove it. please help.

$\lim _{x \to 0^ + } \sin \left( {\frac{1}{x}} \right) = ?$
$\lim _{x \to 0^ + } \sin \left( {\frac{1}{x}} \right) = ?$