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Continuity of Functions

Continuity of Functions

At a Glance - Intermediate Value Theorem

Intermediate Value Theorem (IVT): 

Let f be continuous on a closed interval [a, b]. Pick a y-value M, somewhere between f(a) and f(b)The Intermediate Value Theorem says there has to be some x-value, c, with a < c < b and f(c) = MWe'll use "IVT" interchangeably with Intermediate Value Theorem.  It's just much easier to use an abbreviation.

Here's what's going on, in pictures. Start with a continuous function f on a closed interval [a, b]:

 

Mark on the y-axis where f(a) and f(b) are:
                                                                                      
  
Pick any value of M strictly in between f(a) and f(b):
Draw a horizontal dashed line at height 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:


We've said a continuous function is one we can draw without lifting our pen from the paper. The IVT states this more precisely. If a continuous function starts at 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):

   


If a function on [a, b] skips a value, that function must be discontinuous:


The statement of the theorem may be a little confusing, but with a few pictures this theorem shouldn't be much of a surprise. We're just making the whole notion of "drawing a continuous function without lifting our pencils" a little more formal.

Example 1

Can we use the IVT to conclude that f(x) = sin(x) passes through y = 0 on ?


Example 2

Can we use the IVT to conclude that  passes through y = 1 on ?


Example 3

Can we use the IVT to conclude that  passes through y = 1 on (0, 1)?


Example 4

Can we use the IVT to conclude that f(x) = x2 passes through y = 0 on (-1, 1)?


Exercise 1

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

Exercise 2

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

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