1.4   Continuity

A function f  is said to be continuous at x=a, provided that   and also that (at endpoints, use the appropriate one-sided limit).    In other words, the limit exists and equals the actual value of the function at that limit point.   A function is continuous wherever you can graph it without lifting your pen from the page.   Any places in the graph that require you to lift your pen are places where f  is not continuous (i.e., where f  is discontinuous).  

For all functions continuous at x=a it is true that

                                                 

All of the functions you’ve (probably) ever seen are continuous everywhere, except for possibly a few exceptional points of discontinuity.  So perhaps it’s best to start with examples illustrating some of the ways in which a function can  fail to be continuous at a point.    A function f  is discontinuous at any points where the function is undefined, or where the limit doesn’t exist, or where the function doesn’t equal its limit.    

Click here to see some examples of different types of discontinuity at a point.

Suppose that f  is continuous at all points in an open interval I.  Then f  transforms neighboring points in its domain to neighboring points in its range.  There are no tears or breaks in f’s graph over the interval I.

The following types of functions are continuous at every point in their domains:  polynomials, rational functions, radical functions, trigonometric functions, inverse trigonometric functions, exponential functions, logarithmic functions.  Many of these have points of discontinuity, but these points are not in the function's domain.

Properties of continuous functions    
Suppose that f and g are continuous at x=a, and let c be a constant.   Then the following functions are also continuous at x=a:

f +g,   f – g,    f g,   cf,     f / g   (as long as g(a) is nonzero).

These can be proved using the basic properties of limits.

The composition of continuous functions is also continuous:  
If g is continuous at a and f is continuous at g(a), then  is continuous at a.

The Intermediate Value Theorem
Suppose that f is continuous at every point in the closed interval  [a, b],  then f attains every value between f(a) and f(b).

In other words, to continuously get from here to there, you must pass through all intervening points.

Note that the converse of this theorem is false.   The mid 19^th century text linked to in the introduction shows that this subtlety wasn’t yet generally known back then, and indeed the basic concept of limit hadn’t yet crystallized into it’s current form.   A counter-example is the function

     over the interval  [0, b],   with b>0.

Applications of the intermediate value theorem that may help you to understand it better:  

Assume your speed is a continuous function of time (at all instants of time over some time interval).  Then if you started at 0 mph, and at some point later were going 70 mph, then there had to be some instant in time when you were going at 30 mph (or indeed at any other speed between 0 and 70 mph).  

Suppose that f is continuous at all points in the interval [a, b], and that f(a)<0 and f(b)>0.  Then there must be some point c in the interval (a, b) such that f(c)=0.

Continuity seems a very intuitive concept, and yet has some very surprising consequences.   One is the following fixed-point theorem:

Suppose that  f is continuous at all points in the interval [a, b], and maps points in this interval into this same interval  (e.g., for all x in [a, b],  f(x) is also in [a, b]).    Then there exists a point c in [a, b] such that  f(c) =c.       (Such points are called fixed points)

1.4 Exercises   Do all of them!   :-)