*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 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*.

*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. 3(b) from http://booleweb.ucc.ie/ExamPapers/Exams2008/MathsStds/MS2001a08.pdf

Q. 1(b) from http://booleweb.ucc.ie/ExamPapers/exams2007/Maths_Stds/MS2001Sum2007.pdf

Q. 3(a) from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/MS2001Sum06.pdf

Q. 3(a) from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/Autumn/ms2001Aut.pdf

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

Q. 3(b) from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001Aut05.pdf

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

Q. 2(b) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/MS2001aut.pdf

Q. 2(b) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/MS2001.pdf

Q. 2(b) from http://booleweb.ucc.ie/ExamPapers/exams2001/Maths_studies/MS2001Summer01.pdf

Q. 2(c) from http://booleweb.ucc.ie/ExamPapers/exams/Mathematical_Studies/MS2001.pdf

*From the Class*

1. *Recast the definition of a continuous function (at a point) in terms of –.
*

*2. Prove Proposition 3.3.2 using the Calculus of Limits*

## 2 comments

Comments feed for this article

October 21, 2010 at 10:57 am

Aileen O MahonyJ P,

For exercise sheet 1, Q7 part (iv) do you bring over the underneath inequality and square both side inc the 5?????

Regards

Aileen

October 21, 2010 at 11:08 am

J.P. McCarthyAileen,

That’s exactly what you do.

What makes it possible are the facts that

1. for all , (so you can split the left)

2. (so you can “bring over the underneath” i.e. multiply across by as it is positive)

3. and then . Apply this with and to get (i.e. you can square both sides if both sides are positive)

4. Finally and (you must square the 5 also).

J.P.