ü 475: 4.1: Note on Sectionally Continuous and Sectionally Smooth Functions
First, we make some definitions. For a function f(x), we define the right-hand limit of f at x0 by f x0 + := limhØ0+ f x0 + h
and the left-hand limit of f at x0 by f x0 - := limhØ0- f x0 + h. If both the left- and right-hand limits of f at x0 exist and are
equal, the ordinary limit of f at x0 exists. If both the left- and right-hand limits of f at x0 exist and are not equal, we say f has a
jump discontinuity at x0. If both the left- and right-hand limits of f at x0 exist and are equal, but f is not defined at x0 or
f x0  ∫ f x0 + = f x0 -, we say f has a removable discontinuity at x0 . Such functions can be made continuous by redefining
f x0  to be the common value of the right- and left-hand limits at x0 .
We say a function f(x) is sectionally continuous on an interval [a,b] if and only if f is continuous on [a,b], except possibly for a
finite number of jump or removable discontinuities. A function is sectionally continuous if it is sectionally continuous on any
interval of finite length (in its domain). The graph of a sectionally continuous function has a finite number of removable discontinuities or jumps in any finite interval (in its domain).
We say a function f is sectionally smooth on an interval [a,b] if and only if both f(x) and f '(x) are sectionally continuous on [a,b].
A function is sectionally smooth if it is sectionally smooth on any interval of finite length (in its domain). The graph of a sectionally smooth function has a finite number of removable discontinuities, jumps, or corners in any finite interval in its domain.
Between these points, the graph of f will be continuous, with a continuous derivative. The graph of f will not have any vertical
tangents.
ü Example 1: Some Sectionally Continuous and Sectionally Smooth Functions
ü a. The square wave is sectionally smooth, hence sectionally continuous, but not continuous on [-1,1].
In[1]:=
In[3]:=
fx_ : 1 ; x  0
fx_ :  1 ; x  0
Plotfx, x,  1, 1, Frame  True
1.0
0.5
Out[3]=
0.0
-0.5
-1.0
-1.0
-0.5
0.0
0.5
1.0
2
475SectionallySmoothNote.nb
1
ü b. The function y = x cannot be sectionally continuous on any interval containing the origin. It is sectionally smooth
(and continuous) away from the origin.
In[4]:=
1
Plot , x,  1, 1, Frame  True
x
10
5
Out[4]=
0
-5
-10
-1.0
-0.5
sinx
x
ü c. The function y =
0.0
0.5
1.0
has a removable discontinuity at x = 0. Defining y to be 1 at x = 0, makes the function
continuous. This function is sectionally smooth on any interval.
Sinx
In[5]:=
Plot
x
, x,  2 , 2 , Frame  True
1.0
0.8
0.6
Out[5]=
0.4
0.2
0.0
-0.2
-6
-4
-2
1
0
2
4
6
ü d. h(x) = x 3 is continuous (thus sectionally continuous), but not sectionally smooth in any interval which contains x = 0,
as h'(x) has a vertical tangent at x = 0.
In[6]:=
hx_ : x13 ; x  0
hx_ :   x13 ; x  0
475SectionallySmoothNote.nb
In[8]:=
Plothx, x,  1, 1, Frame  True
1.0
0.5
Out[8]=
0.0
-0.5
-1.0
-1.0
-0.5
0.0
0.5
1.0
3