Exercises


  1. Sketch a possible graph of a function with the given properties.
    1. \( f(3) \) is undefined and \( \displaystyle \lim_{x \rightarrow 3} ~ f(x) = -1 \).
    2. \( f(1) = 3 \) and \( \displaystyle \lim_{x \rightarrow 1} ~ f(x) \) does not exist.
    3. \( f(-1) = 3 \) and \( \displaystyle \lim_{x \rightarrow -1} ~ f(x) = -3 \)
  2. Sketch the graph of a function \( f \) such that \( \displaystyle \lim_{x \rightarrow -1^-} ~ f(x) \) and \( \displaystyle \lim_{x \rightarrow -1^+} ~ f(x) \) both exist and are equal, but \( f \) is discontinuous at \( x = -1 \).
  3. y=-x+4,x<=2;y=x,2<x<4;y=6 at x=4;y=sqrt(4-(x-6))^2+4,4<x<6;y=-(1/4)(x-8)^2+3,x>=6 The graph of \( y = f(x) \) is shown. Determine whether the function is continuous at the indicated points. State the type of discontinuity (removable, jump, infinite, or none of these).
    1. \( x = A \)
    2. \( x = B \)
    3. \( x = C \)
    4. \( x = D \)
    5. \( x = E \)
     
  4. Determine the points at which the following functions are discontinuous. Describe the type(s) of discontinuity at the points from the following list: removable, jump, infinite, or none of these.
    1. \( f(x) = \dfrac{x^2 + x - 2}{x^2 + 3x + 2} \)
    2. \( f(x) = \begin{cases} x + 2 &\text{if } x \leq 3 \\ x &\text{if } x \gt 3 \end{cases} \)
    3. \( f(x) = \dfrac{x + 1}{\left \lvert x - 2 \right \rvert} \)
  5. Find all values of \(x\) for which the given function is continuous.
    1. \( f(x) = 2^{x} \)
    2. \( f(x) = \dfrac{x^{2}+5}{x^{2}-5x} \)
    3. \( f(x) = \dfrac{16x}{x^{2}+16} \)
    4. \( f(x) = \sqrt{2x+3} \)
  6. Sketch the following piecewise functions and determine whether each function is continuous for all real numbers \( x \). Justify your answers.
    1. \( f(x) = \begin{cases} x - 2 &\text{if } x \geq 0 \\ -(x+3) &\text{if } x \lt 0 \end{cases}\)
    2. \(g(x) = \begin{cases} x + 4 &\text{if } x \leq -1 \\ x^{2} - 2x &\text{if } x \gt -1 \end{cases}\)
  7. Your bank account is continuously accruing compounded interest. If you deposit \( $ 500 \) into your account at noon, and the amount of money in your account is plotted against time \( t \), where \( t \) is the number of hours after noon, what type of discontinuity will appear at noon?
    1. Find all values of \( a \) such that the function\[ f(x) = \begin{cases} x^2 - 4x & \text{ if } x \lt a \\ -4 & \text{ if } x \geq a \end{cases} \] is continuous for all values of \( x \).
    2. Find all values of \( a \) and \( b \) such that the function\[ g(x) = \begin{cases} a^2x + 2 &\text{if } x \gt 3 \\ 5 &\text{if } x = 3 \\ x^2 - bx + a &\text{if } x \lt 3 \end{cases} \] is continuous.
  9. For each of the following sets of properties, sketch a function that satisfies the properties given.
    1. \( f(1) = 0 \), \( \displaystyle \lim_{x \rightarrow 1^-} ~ f(x) = -1 \), and \(\displaystyle \lim_{x \rightarrow 1^+} ~ f(x) = 1 \)
    2. \( f(x) = -1 \) for \( -1 \leq x \leq 3 \), \( \displaystyle \lim_{x \rightarrow -1^-} ~ f(x) = 0 \), and \(\displaystyle \lim_{x \rightarrow 3^+} ~ f(x) = 1 \)
  10. For each of the following, sketch the graph of a function \( f(x) \) that satisfies the given description.
    1. \( f \) is continuous for all \( x \neq 2 \), and has a removable discontinuity at \( x=2 \).
    2. The domain of \( f \) is \( \{x \mid 0 \leq x \leq 1, x \in \mathbb{R} \}\), \( f \) is continuous from the right at \(x=0\), continuous on \(0 \lt x \lt 1\), and has an infinite discontinuity at \(x=1\).
    3. \(f\) is continuous for all \(x \neq -1, \frac{1}{2} \), has a removable discontinuity at \( x=-1 \) and an infinite discontinuity at \( x = \frac{1}{2} \).
    4. \(f(2) = -3\), \(f\) is continuous for all \(x \neq 1,3,\) has a jump discontinuity at \(x=3\) and an infinite discontinuity at \(x=1\).
  11. For each of the descriptions in question 10, find an explicit equation of a function \(f(x)\) that satisfies the given description. If possible, try and find a function that is not a piecewise function.