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What is a Continuous Function?

In mathematical analysis, i.e. calculus, continuity is an important concept. What do we mean by a "continuous" function? If we can draw a line or a curve on a piece of paper without lifting our pencil then that line or curve is continuous. In quantitative finance, we study Brownian motion. A Brownian motion - the random zigzagged lines and curves that we observe on our Bloomberg screens when watching the movement of stocks - is mathematically a continuous function (though, it is not "differentiable"). Augustin-Louis Cauchy was the first to give a modern definition of a continuous function. A function is continuous in a given interval if for all values of in that interval "the numerical, that is the absolute, value of the difference of decreases indefinitely with More intuitively, we can say that if the values of a function, , near a point , in an interval do not change abruptly and significantly from the values of at then the function continuous in that interval.


Reference:
  1. Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus by Judith V. Grabiner, Flora Sanborn Pitzer Professor of Mathematics, Pitzer College.
  2. Difference to Differential Equations, An Introduction to Calculus by Dan Sloughter, Furman University

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