f continuous at b, f(b) <> c

[Romeo Colacitti]

If f is continuous at b and f(b) <> c, can I conclude that:

there is an open interval containing b, such that f <> c on that open
interval ?

If yes, why is it so? I have a "gut" feeling, but can't provide myself
a proof.

[G.E. Ivey]

Let epsilon= |f(b)-c|/2

[Google]