Continuity of Functions - Properties of Continuous Functions

Properties of Continuous Functions

Continuous functions are often savvy when it comes to real estate. We've heard they own several properties in East Hampton, Geneva, and Kona. They also hold stake in properties within the Boundedness Theorem, Extreme Value Theorem, and Intermediate Value Theorem. Lucky for us, we're going to visit their properties now—minus the Hamptons, Switzerland, and Hawaii. Sorry for getting your hopes up.

Continuous functions are clean-cut and well groomed compared to other functions. Now's the time for us to talk about what makes them so neat and tidy.

Just a tip: the material in this part may take some time to sink in, so take it slow. We recommend reading things over a couple of times. We also suggest having paper, pencil, and a calculator at hand to draw the functions that arise.

• Boundedness

  The first theorem we'll attack is the boundedness theorem.

  Boundedness Theorem: A continuous function on a closed interval [a, b] must be bounded on that interval.

  What does mean to be bounded again?

  It means there are two numbers—a lower bound M and an upper bound N—such that every value of f on the interval [a, b] falls between M and N. Basically the function can't extend off to ± ∞ on the interval.

  To understand why a function that's continuous on a closed interval [a, b] must be bounded on that interval, now we'll think about what an unbounded function looks like. An unbounded function zooms off to infinity or negative infinity somewhere. 

  If we have a function that is unbounded on a closed interval [a,b] there are only two possibilities. The first possibility is that the function is unbounded at one of the endpoints, in which case the function is discontinuous at that endpoint.

  

  The other possibility is that the function is unbounded somewhere in the middle of the interval, in which case the function is discontinuous somewhere in the middle of the interval.

  

  In either case, an unbounded function on a closed interval [a, b] can't be continuous. Therefore, we can't have a function on a closed interval [a, b] be both continuous and unbounded on that interval. And that means a continuous function on a closed interval [a, b] can't be unbounded (in other words, must be bounded) on that interval.

  These next exercises may require a bit of thought. Try drawing some sample functions.

  The Boundedness Theorem makes two assumptions and draws a conclusion. It says if we assume that both

  •  f is continuous on an interval, and
      
  • that interval is a closed interval, [a,b]

  then we can conclude that f must be bounded on that interval. The examples above illustrate that we do need both assumptions. If either assumption is missing, we're not allowed to draw the conclusion that the function is bounded on the interval.

  We'll say it again. In order to use the Boundedness Theorem to conclude that a function must be bounded on an interval, both of the assumptions must hold.

  If it seems like the answer to the question, "Can we use the Boundedness Theorem?" is usually "no", you would be correct. We need both assumptions in order to use the theorem; if either one fails then we're out of luck.

• Extreme Value Theorem

  We know. The title of this reading sounds pretty gnarly. The extreme value theorem, though, is just a slight extension of the boundedness theorem. There's really nothing all that extreme about it.

  Maximum and Minimum Values

  The maximum value of a function on an interval is the largest value the function takes on within that interval. Similarly, the minimum value of a function on an interval is the smallest value the function takes on within that interval:

  

  A function may hit its maximum and/or minimum value on an interval more than once. The function f(x) = sin(x) on the interval [-2π, 2π] hits its maximum and minimum two times each:

  

  Sample Problem

  Look again at the function f(x) = sin(x) on [-2π,2π]:

  

  The maximum value of the function on this interval is 1. The function attains its maximum at  and at . The minimum value of the function on this interval is -1. The function attains its minimum at  and at . 

  Sample Problem

  Graph the function f(x) = 1 on the interval [0,1]:

  

  The largest value this function hits on the interval is 1, therefore its maximum value is 1. The smallest value this function hits on the interval is 1, so its minimum value is also 1. The function attains both its maximum and its minimum value at every value of x in the interval. How's that for weird?

  Now we're ready to relate the idea of maximums and minimums to continuous functions.

  Extreme Value Theorem: A function f that is continuous on a closed interval [a, b] must attain a maximum and a minimum on that interval.

  To see why this is different from boundedness, look at this function:

  

  This function is bounded, but it never actually reaches a maximum or minimum value. As x approaches ∞ the function is always increasing, approaching N but never quite reaching N. As x approaches -∞ the function is shrinking, approaching M but never quiten reaching it.

  Since a function can be bounded without hitting a maximum or minimum value, the Extreme Value Theorem does say something different from the Boundedness Theorem. In fact, the Extreme Value Theorem is actually a stronger theorem. What we mean by this is that the Boundedness Theorem is really a special case of the Extreme Value Theorem, since a function that attains a maximum and minimum value on a closed interval is also bounded by those values.

  To see why the Extreme Value Theorem makes sense, we'll draw some functions. If we have a continuous function on a closed interval [a,b], it must hit its maximum value in the interval or at an endpoint, as it does here:

  

  

  Similarly, the function must hit its minimum value either in the middle of the interval, or at an endpoint.

  The Extreme Value Theorem makes the same two assumptions as the Boundedness Theorem, but draws a slightly different conclusion. If we assume that both

  •  f is continuous on an interval, and
      
  • that interval is a closed interval [a,b]

  then we can conclude that f hits a minimum and a maximum value on that closed interval.

  Again, if either assumption is missing, we're not allowed to draw the conclusion. If f is discontinuous on [a, b], then f might not hit a maximum or minimum value:

  

  If f is continuous on an interval but the interval is not closed, then f might not hit a maximum or minimum value:

  

  Both assumptions are absolutely necessary.

• 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) = M. We'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.