k Continuity Some functions are nicer than others. The continuous functions are nicely behaved in the sense

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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. 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.