If a partition of an integral diverges, does the whole integral diverge?

[Bipolarity]

$$\int^{b}_{a}f(x)dx = \int^{c}_{a}f(x)dx + \int^{b}_{c}f(x)dx$$

If one of the integrals on the right-hand-side is known to diverge, must the integral on the left also necessarily diverge?

BiP

 

[micromass]

Yes, by definition of improper integrals.

 

[Boorglar]

Yes if c is between a and b. But not necessarily if c is not between a and b:

$$\int_{1}^{2} \frac{1}{x}dx$$ converges, but $$\int_{1}^{-1}\frac{1}{x}dx + \int_{-1}^{2} \frac{1}{x}dx$$ diverges (each term is an integral over a region containing 0, where 1/x is unbounded).