114 CHAPTER 4. CONTINUITY AND LIMITS

8. Show that a real valued function is continuous if and only if it is both upper andlower semicontinuous.

9. Give an example of a lower semicontinuous function which is not continuous and anexample of an upper semicontinuous function which is not continuous.

10. Suppose { fα : α ∈ Λ} is a collection of continuous functions. Define the functionF (x)≡ inf{ fα (x) : α ∈ Λ} . Show F is an upper semicontinuous function. Next letG(x)≡ sup{ fα (x) : α ∈ Λ} . Show G is a lower semicontinuous function.

11. Let f be a function. epi( f ) is defined as {(x,y) : y ≥ f (x)} . It is called the epigraphof f . We say epi( f ) is closed if whenever (xn,yn) ∈ epi( f ) and xn → x and yn → y,it follows (x,y) ∈ epi( f ) . Show f is lower semicontinuous if and only if epi( f ) isclosed. What would be the corresponding result equivalent to upper semicontinuous?

12. Explain why x → exp(sin(ln(x2 +1

)))is continuous on R.

13. Suppose f : N→R is a function. Here N is the set of positive integers. Explain whyf is continuous. Is it necessarily uniformly continuous? Note that you cannot graphthis function without taking pencil off the paper.

4.12 Limit of a FunctionOne of the main reasons for discussing limits of functions is to allow a definition of thederivative. Continuity, derivatives, and integrals are the three main topics in calculus. Sofar, all that has been discussed is continuity. The derivative will be in the next chapter.

In this section, functions will be defined on some nonempty subset of R.

Definition 4.12.1 A point x is a limit point of a nonempty set D means that B(x,δ )always contains a point of D different than x for any δ > 0. Then if D is the domain of afunction f , and L ∈ R, we say that limy→x f (y) = L means: For every ε > 0, there exists aδ > 0 such that whenever y ∈ D and 0 < |y− x|< δ , it follows that | f (y)−L|< ε.

If x is a limit point of D+ ≡ D∩ (x,∞) , then limy→x+ f (y) = L means the same thingexcept y is restricted to D+. If x is a limit point of D− ≡D∩(−∞,x) , then limy→x− f (y)= Lmeans the same thing except you restrict y to D−.

Limits are also taken as a variable “approaches” infinity. Of course nothing is “close”to infinity and so this requires a slightly different definition. Suppose D contains all xsufficiently large. Then limx→∞ f (x) = L if for every ε > 0 there exists l such that wheneverx > l, | f (x)−L| < ε and limx→−∞ f (x) = L if for every ε > 0 there exists l such thatwhenever x < l, | f (x)−L|< ε holds.

The main example of interest in this book is when the limit point is either the interiorof an interval, the end point of an interval or an end point of two adjacent intervals.

The following pictures illustrate some of these definitions.