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Continuity of Functions

Continuity of Functions

Intermediate Value Theorem Exercises

Example 1

  • Can we use the IVT to conclude that f(x) = x3 + 2x + 1 passes through y = 0 on the interval (-2, 2)?

Example 2

  • Can we use the IVT to conclude that f(x) = ex 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) = x2 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?