The spin-1/2 chain with isotropic Heisenberg exchange $J_1$, $J_2 > 0$ between first and second neighbors is frustrated for either sign of J1. Its quantum phase diagram has critical points at fixed $J_1/J_2$ between gapless phases with nondegenerate ground state (GS) and quasi-long-range order (QLRO) and gapped phases with doubly degenerate GS and spin correlation functions of finite range. In finite chains, exact diagonalization (ED) estimates critical points as level crossing of excited states. GS spin correlations enter in the spin structure factor $S(q)$ that diverges at wave vector $q_m$ in QLRO($q_m$) phases with periodicity $2pi/q_m$ but remains finite in gapped phases. $S(q_m)$ is evaluated using ED and density matrix renormalization group (DMRG) calculations. Level crossing and the magnitude of $S(q_m)$ are independent and complementary probes of quantum phases, based respectively on excited and ground states. Both indicate a gapless QLRO($pi/2$) phase between $-1.2 < J_1/|J_2| < 0.45$. Numerical results and field theory agree well for quantum critical points at small frustration $J_2$ but disagree in the sector of weak exchange $J_1$ between Heisenberg antiferromagnetic chains on sublattices of odd and even-numbered sites.
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