Is it always true that an integral diverges on (a,b) if it diverges on (c,b) and a

Question

I know that the integral of 1/x diverges on (0,b). I know the same integral still diverges on (-b, b), despite the fact it seems it should be zero. My question then is, "Is it ALWAYS the case that if an integral diverges on a subinterval of an interval, it must then diverge on the larger interval?".