Continuity · Definition

Intermediate value theorem

calculuscontinuitytheorems

Before you start you need to know

This theorem formalizes an intuitive idea: a continuous function cannot move from one value to another without taking every value in between.

Definition

If \(f\) is continuous on \([a,b]\) and \(N\) lies between \(f(a)\) and \(f(b)\), there is some \(c\in(a,b)\) with \(f(c)=N\).

\[f\ \text{continuous on }[a,b],\ f(a)

If you draw a continuous curve that starts below the axis and ends above it, it must cross it at some point. The intermediate value theorem says exactly that, for any level between \(f(a)\) and \(f(b)\).

It is an existence theorem: it guarantees a solution to \(f(x)=N\) but not where it is. That is why it is used to prove an equation has a root and, with bisection, to approximate it.

Continuity is essential: a function with a jump can "skip" values.

Think about it

Show that \(x^3+x-1=0\) has a root in [0,1].

The function is continuous, f(0)=−1<0 and f(1)=1>0; by the intermediate value theorem there is c in (0,1) with f(c)=0.

Prove that the equation \(\cos x=x\) has a solution in \([0,1]\).

Let \(h(x)=\cos x-x\), which is continuous. Then \(h(0)=1>0\) and \(h(1)=\cos1-1<0\), so there is \(c\in(0,1)\) with \(h(c)=0\), i.e. \(\cos c=c\).

Exercises 20

  1. Let \(f(x)=x^{3}+x-1\). On which interval does the intermediate value theorem guarantee a root?
  2. Let \(f(x)=x^{2}-3\). On which interval does the intermediate value theorem guarantee a root?
  3. Let \(f(x)=\cos x-x\). On which interval does the intermediate value theorem guarantee a root?
  4. Which hypothesis on \(f\) does the intermediate value theorem require on \([a,b]\)?
  5. If \(N\) lies between \(f(a)\) and \(f(b)\), what does the intermediate value theorem state?
  6. Let \(f(x)=x^{3}-x-1\). On which interval does the intermediate value theorem guarantee a root?
  7. Let \(f\) be continuous on \([0,4]\) with \(f(0)=2\) and \(f(4)=-1\). What can be guaranteed?
  8. Let \(f(x)=1/x\) on \([-1,1]\). Can the theorem be used to find \(c\) with \(f(c)=0\)?
  9. What kind of result is the intermediate value theorem?
  10. Let \(f\) be continuous on \([1,3]\) with \(f(1)=4\) and \(f(3)=-2\). Which statement is guaranteed?
  11. Let \(f(x)=x^3-x-1\) on \([1,2]\). Since \(f(1)=-1\) and \(f(2)=5\), what does the intermediate value theorem guarantee?
  12. For \(f(x)=x^2\) on \([0,3]\), which \(c\in[0,3]\) satisfies \(f(c)=5\)?
  13. Let \(f(x)=\cos x\) on \([0,\pi]\). For which value \(y\) does the theorem guarantee some \(c\) with \(f(c)=y\)?
  14. Let \(f(x)=x^2-3\) on \([1,2]\). After evaluating the midpoint \(1.5\), on which interval is a root guaranteed?
  15. Why can the theorem not be used to ensure a zero of \(f(x)=1/x\) on \([-1,1]\), even though \(f(-1)=-1\) and \(f(1)=1\)?
  16. Let \(g\) be continuous on \([0,1]\) with \(0\le g(x)\le1\). To prove that \(g(c)=c\) for some \(c\), to which function is the theorem applied?
  17. For \(p(x)=x^3+x+1\), on which interval does the theorem ensure a root?
  18. Let \(f\) be continuous on \([0,1]\) with \(f(0)=2\) and \(f(1)=5\). Which statement must be true?
  19. Let \(f(x)=\lfloor x\rfloor\) (floor function) on \([0,2]\), with \(f(0)=0\) and \(f(2)=2\). It never equals \(\tfrac12\). Why does this not contradict the theorem?
  20. How many real roots does \(f(x)=x^3-3x\) have on \([-2,2]\), where \(f(-2)=-2\) and \(f(2)=2\)?

What comes next

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. §4.5, Teorema 4.5.1
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 4, Teorema 4.23
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §5.3, Teorema 5.3.7