Recall that a function f is said to be continuous at a point "a" if three conditions are satisfied:
1) f(a) is defined
2) 
3) 
Recall also that 
exists if and
only if both the right hand limits and left-hand exist and are equal. That is
if and only if
I. Using this as background we would like you now to
discover the behavior of the following functions. You should do this by finding the following for each
function:
a) f(a)
b) 
c) 
d) 
You also should ask yourself the following questions about each function:
i) Is the function continuous or discontinuous at "a"?
ii) If the function is discontinuous at "a" what is the nature of that discontinuity?
iii) What is it about the function that causes the discontinuity at this point?
NOTE: Review the "piecewise" command in the second orientation lab; this command is needed here.
For the next 2 functions, first graph by hand then proceed as in the
previous 7 problems.
Now summarize what you've learned above and write your summary in paragraph form. State what types of discontinuities there are. In what types of functions do these discontinuities occur? For specific types of functions where do you look for discontinuities? What can you say about points of continuity?
II. Construct functions defined at all real
numbers that have the following properties.
Give an explicit (non-graphical) description of the function.
J. An infinite discontinuity at x=2, a removable discontinuity at x=5, continuous at all other real numbers, and name J.
K. Jump
discontinuities at
, an infinite discontinuity at x=1, continuous at all other real numbers, and name K.
L.
L is continuous at x=1 and L is discontinuous at x=2.
M. 
III. OPTIONAL PROBLEM
Let
and
Discuss points of continuity and discontinuity for these two functions.
Mathematics and Computer Science Department, The College of Wooster, Wooster, OH 44691