right limit of a continuous bounded function
i was thinking whether the following is true or false:
Let : be a continuous and bounded function. then, has a right limit at .(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 served as a counter example as the claim would be violated at .
intuitively it seems that the right limit at should exist but i am not able to prove it. please help.