Continuity · Definition

Continuous function

calculuscontinuity

Before you start you need to know

Continuity describes functions whose graph has no jumps, holes or asymptotes at a point. It is a basic idea of calculus.

Definition

A function \(f\) is continuous at \(a\) if \(f(a)\) is defined, its limit as \(x\) tends to \(a\) exists and equals \(f(a)\).

\[\lim_{x\to a}f(x)=f(a)\]

Intuitively, a function is continuous if you can draw its graph without lifting the pencil. The definition makes it precise with three conditions: \(f(a)\) exists, the limit exists, and the two agree.

Guide example: let \(f(x)=\dfrac{x^{2}-4}{x-2}\). It is not defined at \(x=2\), so it is not continuous there. However, its limit at 2 is 4; defining \(f(2)=4\) makes the function continuous.

Typical mistake: thinking that \(f(a)\) being defined is enough. If the limit does not exist or differs from \(f(a)\), the function is not continuous at \(a\), even though the value at the point exists.

Think about it

Which three conditions are required for continuity at a?

That f(a) is defined, that the limit at a exists, and that the limit equals f(a).

Prove from the definition that \(f(x)=x^2\) is continuous at \(c=2\).

If \(|x-2|<\delta\le1\), then \(|x+2|<5\) and \(|x^2-4|<5\delta\). With \(\delta=\min\{1,\varepsilon/5\}\) we get \(|x^2-4|<\varepsilon\).

Exercises 20

  1. Which of the following is NOT a condition for f to be continuous at a?
  2. At which point is the function \(f(x)=\dfrac{1}{x-4}\) discontinuous?
  3. Which value of k makes \(f(x)=\begin{cases}\dfrac{x^{2}-4}{x-2}&\text{if }x\neq2\[2mm] k&\text{if }x=2\end{cases}\) continuous at \(x = 2\)?
  4. Which value of k makes \(f(x)=\begin{cases}x+1&\text{if }x<1\\ kx^{2}&\text{if }x\geq1\end{cases}\) continuous at \(x = 1\)?
  5. Which value of k makes \(f(x)=\begin{cases}kx-3&\text{if }x\leq2\\ x^{2}-1&\text{if }x>2\end{cases}\) continuous at \(x = 2\)?
  6. Which value of k makes \(f(x)=\dfrac{x^{2}-9}{x-3}\) continuous at \(x = 3\) when \(x\neq3\) and \(f(3)=k\)?
  7. Which value of k makes continuous at \(x = 1\) the function defined by \(f(x)=x+k\) if \(x<1\) and \(f(x)=2x^{2}\) if \(x\geq1\)?
  8. Which statement about the continuity of \(f\) at a point \(a\) is correct?
  9. What type of discontinuity does \(f(x)=\dfrac{x^{2}-1}{x-1}\) have at \(x = 1\)?
  10. What is the value of \(\lim_{x\to3}(2x^{2}-x)\)?
  11. For \(f(x)=2x-5\) at \(c=3\) and \(\varepsilon=0.4\), what is the largest \(\delta\) that works?
  12. Let \(f(x)=\dfrac{x^2-4}{x-2}\) if \(x\ne2\) and \(f(2)=k\). For which \(k\) is it continuous at \(2\)?
  13. Which of these functions is continuous at \(0\)?
  14. The Dirichlet function equals \(1\) at rationals and \(0\) at irrationals. At how many points is it continuous?
  15. For \(f(x)=x^2\) at \(c=1\) and \(\varepsilon=0.05\), what is the largest \(\delta\) that works?
  16. Find \(a\) so that the function is continuous at \(2\): \(f(x)=ax+1\) if \(x\le2\), \(f(x)=x^2-ax\) if \(x>2\).
  17. If \(f\) and \(g\) are continuous at \(c\), which of these functions is always continuous at \(c\)?
  18. Let \(f(x)=x\sin(1/x)\) for \(x\ne0\) and \(f(0)=k\). For which \(k\) is it continuous at \(0\)?
  19. If \(f\) is continuous at \(c\) and \(f(c)>0\), what can be guaranteed?
  20. For \(f(x)=1/x\) at \(c=1\), if \(|x-1|<\delta\le1/2\), what is the smallest bound of the form \(k\delta\) valid for \(|f(x)-f(1)|\)?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.
  • Michael Spivak, Calculus, 4.ª edición, Publish or Perish, 2008.
  • Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015. §4.3, Definición 4.3.1
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 4, Definición 4.5
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §5.1, Definición 5.1.1