RealClimate.org has published a very informative article, "Extremely Hot" by Stefan Rahmstorf and Dim Coumou that explores the impacts of even modest one or two degree C warming on severe climate events such as heatwaves.
The article uses these graphs to demonstrate that the impacts of even modest warming create non-linear repercussions. This is part of what is often described as shifting the world's climate into a new steady state.
What you see is that the mild or mid-range between very cold and very hot diminishes to produce somewhat fewer very cold events and substantially more very hot and record hot events. And that's where we're heading if we don't soon take meaningful action to arrest global warming.