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Continuity Some functions are nicer than others. The continuous functions are nicely behaved in the sense of pointwise limits. Definition 10 (Continuity) We say that a function f is continuous at a point r = a provided that: (a) The function value f(a) exists; (b) The limit lima f(x) exists (as a finite number); (c) These are both the same number, i.e. lim f(x) = f(a). D+I In other words, a function y = f(x) is continuous when small changes in a produce small changes in y. Notice that points of discontinuity occur when the function is either undefined, or where the limit does not exist. A function may also be discontinuous when the limit and the function disagree (have different values). 20. Sketch a graph of the greatest integer function f(x)= []. This function is also known as the floor function. You can obtain this graph in Desmos by typing "y = floor(x)". (a) What is the limit of this function as r approaches 1 from the left? In other words, what is lim [2]? 2-1- (b) What is the limit of this function as r approaches 1 from the right? In other words, what is lim [2]? x+1+ (c) What can you say about lim[2]? (d) Is the floor function continuous at x 1? Explain your answer. (e) Where is floor function discontinuous? Explain your answer. (f) Where is the floor function is continuous? Explain your answer.