- Let f= 2 x + 7. Show f is continuous at every point x. Hint: You need to let ε > 0 be given. In this case, you should try δ ≤ ε∕2. Note that if one δ works in the definition, then so does any smaller δ.
- Suppose D=∪and f= x onwhile f= 5 . Is f continuous at the point, 9? Use whichever definition of continuity you like.
- Let f= x
^{2}+ 1. Show f is continuous at x = 3. Hint:= = .Thus if

< 1, it follows from the triangle inequality,< 1 + 3 = 4 and soNow try to complete the argument by letting δ ≤ min

. The symbol, min means to take the minimum of the two numbers in the parenthesis. - Let f= 2 x
^{2}+ 1. Show f is continuous at x = 1. - Let f= x
^{2}+ 2x. Show f is continuous at x = 2. Then show it is continuous at every point. - Let f=. Show f is continuous at every point. Hint: Review the two versions of the triangle inequality for absolute values.
- Let f=. Show f is continuous at every value of x.
- If x ∈ ℝ, show there exists a sequence of rational numbers, such that x
_{n}→ x and a sequence of irrational numbers,such that x_{n}^{′}→ x. Now consider the following function.Show using the sequential version of continuity in Theorem 6.1.1 that f is discontinuous at every point.

- If x ∈ ℝ, show there exists a sequence of rational numbers, such that x
_{n}→ x and a sequence of irrational numbers,such that x_{n}^{′}→ x. Now consider the following function.Show using the sequential version of continuity in Theorem 6.1.1 that f is continuous at 0 and nowhere else.

- Suppose y is irrational and y
_{n}→ y where y_{n}is rational. Say y_{n}= p_{n}∕q_{n}. Show that lim_{n→∞}q_{n}= ∞. Now consider the functionShow that f is continuous at each irrational number and discontinuous at every nonzero rational number.

- Suppose f is a function defined on ℝ. Define
Note that these are decreasing in δ. Let

Explain why f is continuous at x if and only if ωf

= 0. Next show thatNow show that ∪

_{n=1}^{∞}is an open set. Explain why the set of points where f is continuous must always be a G_{δ}set. Recall that a G_{δ}set is the countable intersection of open sets. - Show that the set of rational numbers is not a G
_{δ}set. That is, there is no sequence of open sets whose intersection is the rational numbers. Extend to show that no countable dense set can be G_{δ}. Using Problem 11, show that there is no function which is continuous at a countable dense set of numbers but discontinuous at every other number. - Use the sequential definition of continuity described above to give an easy proof of Theorem 6.0.6.
- Let f=show f is continuous at every value of x in its domain. For now, assumeexists for all positive x. Hint: You might want to make use of the identity,
at some point in your argument.

- Using Theorem 6.0.6, show all polynomials are continuous and that a rational function
is continuous at every point of its domain. Hint: First show the function given as
f= x is continuous and then use the Theorem 6.0.6. What about the case where x can be in F? Does the same conclusion hold?
- Let f=and consider g= f. Determine where g is continuous and explain your answer.
- Suppose f is any function whose domain is the integers. Thus D= ℤ, the set of whole numbers,,−3,−2,−1,0,1,2,3,. Then f is continuous. Why? Hint: In the definition of continuity, what if you let δ =? Would this δ work for a given ε > 0? This shows that the idea that a continuous function is one for which you can draw the graph without taking the pencil off the paper is a lot of nonsense.
- Give an example of a function f which is not continuous at some point but is continuous at that point.
- Find two functions which fail to be continuous but whose product is continuous.
- Find two functions which fail to be continuous but whose sum is continuous.
- Find two functions which fail to be continuous but whose quotient is continuous.
- Suppose f is a function defined on ℝ and f is continuous at 0. Suppose also that
f= f+ f. Show that if this is so, then f must be continuous at every value of x ∈ ℝ. Next show that for every rational number, r, f= rf. Finally explain why f= rffor every r a real number. Hint: To do this last part, you need to use the density of the rational numbers and continuity of f.
- Show that if r is an irrational number and → r where p
_{n},q_{n}are positive integers, then it must be that p_{n}→∞ and q_{n}→∞. Hint: If not, extract a convergent subsequence for p_{n}and q_{n}argue that to which these converge must be integers. Hence r would end up being rational.

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