right limit of a continuous bounded function

i was thinking whether the following is true or false:

Let $\displaystyle f$:$\displaystyle (a,b)$ $\displaystyle \rightarrow \mathbb{R}$ be a continuous and bounded function. then, $\displaystyle f$ has a right limit at $\displaystyle a$.(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 $\displaystyle sin(1/x)$ served as a counter example as the claim would be violated at $\displaystyle x=0$.

intuitively it seems that the right limit at $\displaystyle a$ should exist but i am not able to prove it. please help.