On Monday we again wrote down the definition of a continuous function (at a point). We showed that if we combine continuous functions in ‘nice’ ways they stay continuous. We showed as a corollary that polynomials are continuous everywhere and we stated without proof that $\sin x,\,\cos x$ are also continuous. We defined the composition of two functions and said that, under composition, continuous functions remain continuous (proof on this page). We did a few examples – showing how a function can fail to be continuous by not being defined, by not having a limit, or by the limit not being equal to the value of a function at a point. Finally we said if a function had a discontinuity at a point, if a redefinition of the function at that point could make the function continuous, then the discontinuity is called removable, otherwise it is called essential.
In the tutorial we went through the Sample test. We also did from Ex.Sh.1 Q 7. (vi). Someone asked about Q. 8 (iii) – we didn’t have time to finish it in class but I have the solution on this page.
Finally, for Q. 3 of the test, you need to know the following definitions:
even, odd, increasing, decreasing, quadratic, roots, polynomial, rational function, absolute value, limit, one-sided limit, continuous at a point, continuous, composition

Problems

You need to do exercises – all of the following you should be able to attempt. Do as many as you can/ want in the following order of most beneficial:

Wills’ Exercise Sheets

Q. 1 (ix),  6,  7, 8 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise2.pdf

Past Exam Papers

Q. 2(b), 3(a) from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001.pdf

Q. 1(b), 3(b) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/ms2001s2004.pdf

From the Class

1. Recast the definition of a continuous function (at a point) in terms of $\varepsilon$ $\delta$.

2. Prove Proposition 3.3.2 using the Calculus of Limits