- Topics At a Glance
- Continuity at a Point
- Continuity at a Point via Pictures
- Continuity at a Point via Formulas
- Functions and Combinations of Functions
- Continuity on an Interval
- Continuity on an Interval via Pictures
- Continuity on an Interval via Formulas
- Continuity on Closed and Half-Closed Intervals
- Determining Continuity
- The Informal Version
- The Formal Version
**Properties of Continuous Functions**- Boundedness
- Extreme Value Theorem
**Intermediate Value Theorem**- In the Real World
- Page: I Like Abstract Stuff; Why Should I Care?
- Page: How to Solve a Math Problem
- Appendix: Intervals and Interval Notation

**Intermediate Value Theorem (IVT):**

Let *f* be continuous on a closed interval [*a*,*b*]. Pick a *y*-value *M* with *f*(*a*)*b*) or *f*(*b*)*a*). *x*-value *c* with *a* < *c* < *b* and *f*(*c*) = *M*

*f* on a closed interval [*a*,*b*]:

*y*-axis where *f*(*a*) and *f*(*b*) are:

*M* strictly in between *f*(*a*) and *f*(*b**y* = *M*. The IVT guarantees that this dashed line will hit the graph of *f*. In other words, the IVT guarantees the existence of some value *c* strictly in between *a* and *b* where the function value is *M**f*(*a*) and ends at at *f*(*b*), then as *x* travels from *a* to *b* the function must hit every *y* value in between *f*(*a*) and *f*(*b*): *a,b*] skips a value, that function must be discontinuous:

Example 1

Can we use the IVT to conclude that |

Example 2

Can we use the IVT to conclude that passes through |

Example 3

Can we use the IVT to conclude that passes through |

Example 4

Can we use the IVT to conclude that |

Exercise 1

- Can we use the IVT to conclude that
*f*(*x*) =*x*^{3}+ 2*x*+ 1 passes through*y*= 0 on the interval (-2,2)?

Exercise 2

- Can we use the IVT to conclude that
*f*(*x*) =*e*^{x}passes through y = 0.1 on the interval (0,1)?

Exercise 3

- Can we use the IVT to conclude that
*f*(*x*) = sin(*x*) equals 0.4 at some place in the interval ?

Exercise 4

- Can we use the IVT to conclude that
*f*(*x*) = tan(*x*) equals 0 for some*c*in (0,π)?

Exercise 5

- Can we use the IVT to conclude that
*f*(*x*) =*x*^{2}passes through 1 on the interval (-1,1)?

Exercise 6

- Draw a function that is continuous on [0,1] with
*f*(0) = 0,*f*(1) = 1, and*f*(0.5) = 20.

Exercise 7

- Suppose that
*f*hits every value between*y*= 0 and*y*= 1 on the interval [0,1]. Must*f*be continuous on that interval?