Thom uses the (originally Greek) word "catastrophe" in a technical sense (thinking of the meaning "turn-around" or "sudden change", rather than the modern meaning "disaster").

We can illustrate the idea by taking the one form of elementary catastrophe which exists in two-dimensional space. From a topological point of view, any continuous deformation of a plane is continuous. But a discontinuity can be introduced by allowing a fold (and some functions have such discontinuities). Click for some illustrations.



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