Israel resumed air strikes in Gaza after Palestinian militants fired rockets following the end of a three-day truce on Friday morning.
Five people were killed in Gaza, while two Israelis were injured by mortars.
The Israeli army called the renewed rocket attacks "unacceptable, intolerable and short-sighted".
The Palestinian militant group Hamas, which dominates Gaza, earlier rejected any extension of the truce, saying Israel had failed to meet its demands.
Israeli government officials said they had pulled out of Egyptian-brokered negotiations with Hamas and the other Palestinian factions in Cairo, stressing they would not "negotiate under fire".
But Egypt called on both sides to return to the negotiating table.
At the scene: Wyre Davies, Kerem Shalom in southern Israel
The border area didn't feel particularly safe this morning, and several kibbutzim were still virtual ghost towns as more than 36 rockets and mortars were fired from Gaza.
But that, perhaps, is part of the Hamas strategy. They don't want Israelis to feel safe or comfortable as long as the blockade of Gaza continues.
While Gaza's 1.8 million inhabitants live, penned in, barely able to make a living - so the argument goes - why should Israelis just across the fence feel any more secure?
On Friday, the Israel Defense Forces (IDF) resumed its offensive with aircraft, tanks and gunboats, attacking some 35 targets.
A 10-year-old boy was the first to die, when a missile struck near a mosque in Gaza City, Palestinian health ministry officials in Gaza said.
Another three people were killed in central Gaza while an Islamic Jihad militant died in Rafah, they added.
The IDF earlier announced it was striking "terror sites across the Gaza Strip" in response to renewed rocket fire.
Militants began firing missiles from Gaza shortly before the ceasefire ended at 08:00 local time (05:00 GMT).
By Friday afternoon, the IDF said more than 52 rockets and mortars had been fired at Israel, injuring two Israelis.
Israel's Iron Dome missile defence system had intercepted three rockets and 38 had landed in Israel, it added.
Human cost of the conflict
- 1,922 killed, including at least 1,407 civilians
- 448 children
- 235 women
- 64 soldiers
- 2 civilians
- 1 Thai national in Israel
(Source: OCHA; 0500 GMT on 8 August)
The violence resumed after Israeli and Palestinian representatives failed to agree a long-term truce at indirect talks in the Egyptian capital, Cairo.
Hamas said Israel had failed to meet its key demands, including the lifting of the blockade of Gaza and the freeing of about 100 prisoners released in exchange for Gilad Shalit in 2011 who have been re-arrested.
The group also rejected Israel's call for the demilitarisation of Gaza. A spokesman warned the militant group was ready for "a long war".
However, Hamas said the Palestinian factions were willing to continue the talks despite the fresh violence.
The Israeli delegation left Cairo on Friday morning.
"Hamas is now continuing to fire towards Israeli civilians, while Israel fully respected the ceasefire and announced it is ready to extend it," a senior official told the BBC.
On Friday afternoon, Egypt's foreign ministry called for an immediate resumption of negotiations on a long-term truce, stressing that only a few outstanding issues remained.
"The foreign ministry calls on all sides to rise to their responsibilities... and to return immediately to the ceasefire commitment and exploit the opportunity available to resume negotiations on the very limited sticking points that remain in the fastest possible time," a statement said.
At least 1,922 Palestinians, mostly civilians, have died, according to latest United Nations figures.
Israel's government says 64 soldiers have been killed, along with two Israeli civilians and a Thai national. It also claims that about 900 Palestinian militants have died in the violence.
Israel launched Operation Protective Edge on 8 July with the stated aim of halting rocket fire from militants in Gaza and destroying the network of tunnels it said were used by militants to launch attacks inside Israel.
On Thursday, the human rights group Amnesty International called for an investigation into what it said was mounting evidence that Israeli forces had deliberately attacked hospitals and health professionals in Gaza. The attacks have left at least six medics dead.