Yeddyurappa has accused the chief minister of being "arrogant."
The BJP has decided to corner the government on a host of issues, both inside and outside the assembly during the ongoing legislature session.
"His (Yeddyurappa's) comments are not worth replying. I do not need Yeddyurappa's certificate," Kumaraswamy told reporters in response to the remarks of the state BJP chief.
Both Kumaraswamy and Yeddyurappa indulged in a verbal duel, even as the Karnataka legislative assembly met for a ten-day long winter session in this border city.
The assembly was adjourned for the day after obituary references to former Prime Minister Atal Bihari Vajpayee, Union Minister Ananth Kumar, senior Congress leader C K Jaffer Sharief, actor Ambareesh and other former MLAs who died recently.
Reacting to the BJP and Yeddyurappa's threat about agitation both inside and outside the House against his government on several issues, including the plight of farmers, the chief minister said, it was their duty.
"I welcome their agitation......we are ready to respond (in the House)," he added.
Earlier in the day, Yeddyurappa accused Kumaraswamy of being "arrogant."
He alleged that the chief minister was speaking lightly about the opposition and the media.
"He (Kumaraswamy) thinks opposition has to work like his slaves....and media to work according to him. This will not happen," Yeddyurappa said.
Yeddyurappa accused Kumaraswamy of neglecting north Karnataka and lack of clarity on farm loan waiver.
"Your drama will not work.....you do not have moral right to continue in power, quit immediately " Yeddyurappa said.
"Farmers are protesting fed up with your government's failures and BJP will continue its agitation against the government both inside and outside the House, and will continue its agitation throughout the state after the session," he added.
The BJP's core committee that met Monday discussed the strategy to be adopted to corner the coalition government during the winter session.
BJP is likely to move an adjournment motion in the assembly on agrarian crisis in the state, according to party sources.