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.