a function f(x) is continuous at the closed interval or bounded only if:
f(x) is continuous for all values of x in the interval (a,b) and
f(x) is right continuous at f(a)
and left continuous at f(b).
In the above question f(0) doesn't exist hence the function is not bounded in the closed interval [a,b]. So its definitely not a bounded function.
Hope that answers the doubt on how to know whether a function is bounded or not.