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# Continuity of Functions

# Intermediate Value Theorem Exercises

### Example 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)?

### Example 2

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

### Example 3

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

### Example 4

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

### Example 5

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

### Example 6

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

### Example 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?