The bombardment on Saturday came after Hamas, which rules Gaza, and the Islamic Jihad movement fired more than 200 rockets towards cities and villages in the occupied territories.
At least three Palestinians, including the mother, infant and a 22-year-old male, Emad Naseer, were killed in the air raids, the health ministry in Gaza said, while 13 others were wounded.
The escalation began on Friday when an Israeli air strike killed two fighters from Hamas.
Two other Palestinians protesting near the frontier were also killed by Israeli forces, Al Jazeera reported.
Hamas and Islamic Jihad responded with rocket fire on Saturday.
In a statement, the two groups promised a "broader and more painful" response if Israel "pursues its aggression". Israeli forces retaliated with air raids and tank fire against more than 30 targets belonging to both groups.
Explosions in Gaza City, where busy streets were packed with shoppers making preparations for the holy Muslim month of Ramadan, shook buildings and sent people fleeing for cover.
Ibtessam Abu Arar, aunt of the 14-month old infant who died in the Israeli raid, said, “The Israeli plane fired a missile near the house and the shrapnel entered the house and hit the poor baby."
The child's mother, 37-year-old Falestine Abu Arar, died from her wounds hours later, the health ministry said in a statement.
Spokesman for the al-Aqsa Martyrs Brigade, a military wing of Fatah movement, said the martyrdom of Palestinian people in the Israeli attacks would only strengthen the resolve of fighters in defensing Palestine.
Israel and Egypt have maintained a crippling blockade on Gaza since Hamas seized control of the territory in 2007.
Following heavy fighting in late March, Israel agreed to ease the blockade in exchange for a halt to rocket fire. This included expanding a fishing zone off Gaza's coast, increasing imports into Gaza and allowing the Persian Gulf state of Qatar to deliver aid to the cash-strapped territory.
But Israel scaled back the fishing zone this week in response to rocket fire and shut the border crossings entirely on Saturday after barrages from the enclave.
Some 2 million Palestinians live in Gaza, whose economy has suffered years of blockades as well as recent foreign aid cuts. Unemployment stands at 52 percent, according to the World Bank, and poverty is rampant.
Israel says its blockade is necessary to stop weapons reaching Hamas, with which it has fought three wars since the group seized control of Gaza in 2007.
Hamas said on Thursday that its Gaza chief, Yeyha Sinwar, had travelled to Cairo for talks on efforts to maintain calm along the border and alleviate hardship in the enclave.
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