This week in Calculus, we learned about continuity. This is a topic that has been covered before in pre-calculus so it was mainly a review and a little more in depth information.
There was an interesting problem on in one of our lab assignments dealing with the continuity of salt and pepper functions that I had a hard time with at first. A picture of the problem is below. When I graphed it, I could see that there where infinite places where it was continuous but I didn't initially know how to define those points. After talking with my peers I started to figure it out. To find where the function is continuous, you have to find the x values at which both the rational and irrational sides of the function are approaching the same y value. Since the rational side is always approaching 0, you must find where sin(π/x) equals zero. I like to think of it in terms of the unit circle. Looking at a sin(x) curve, sin(x)= 0 where x =πk. Since the problem calls for π/x not x, we are looking at where π/x =πk. I solved for x and found that x= 1/k where k is any integer.
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