Continuity & Differentiability

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Part A: Continuity Exercises for This Topic Index of OnLine Topics Everything for Calculus Everything for Finite Math Everything for Finite Math & Calculus Utility: Function Evaluator & Grapher 
Part B: Differentiability
Note To understand this topic, you will need to be familiar with derivatives and limits, as discussed in the chapter on the subject in Calculus Applied to the Real World. If you like, you can review the topic summary material on derivatives and limits or, for a more detailed study, the online tutorials on derivatives and limits.
To begin, we recall the definition of the derivative of a function, and what it means for a function to be differentiable.
Derivative; Differentiability
The derivative of the function f at the point a in its domain is given by
We say that function f is differentiable at the point a in its domain if f'(a) exists. Differentiable on a Subset of the Domain
Note

_{}
Example 1 Functions Not Differentiable at Isolated Points
Determine points of nondifferentiability of the following functions
(a)  f(x)  =  (x1)^{1/3}  (b)  g(x)  =  x+2  (c)  r(x)  = 
Solution
(a) The power rule tells us that f(x) = (x1)^{1/3} has derivative f'(x) = (1/3)(x1)^{2/3} everywhere where this expression is defined, and is not diffeentiable when (1/3)(x1)^{2/3} is not defined. Since (x1) has a negative exponent, f'(x) is not defined when x = 1, and so f is not differentible there. In fact, direct calculation shows that
lim h0 
h 
=  lim h0 
h 
=  lim h0 
h^{2/3} 
=  +, 
(b)  Since g(x) = x+2 =  (x+2) x+2  if x 2 if x > 2 
, 
lim h0 
h 
=  lim h0 
h  . 
(c) The quotient rule tells us that r(x) = x^{2}/(x  1) is differentiable everywhere except at x = 1. However, x = 1 is not in the domain of r, and so r is differentiable at every point of its domain.
As we see in the graph on the right, there are no points of vertical tangency or cusps.
Before We Go On...
As you can see, the graphs provide immediate information as to where to look for a point of nondifferentiability: a point where there appears to be a cusp or a vertical tangent.
Here is one for you.
_{} Example 2 Points of NonDfferentiability
Q In Part A we discussed continuity, and here we discussed differentiability. Are all continuous functions differentiable? Are all differentiable functions continuous?
A Briefly:
(a) Not all continuous functions are differentiable. For instacne, the closedform function f(x) = x is continuous at every real number (including x = 0), but not differentiable at x = 0.
(b) However, every differentiable function is continuous. More precisely, we have the following theorem.
_{} Theorem Differentiability Implies Continuity
If f is differentiable at a, then it is continuous at a.
Proof
Suppose that f is differentiable at the point x = a. Then we know that
lim h0 
h 
exists, and equals f'(a). 
lim h0 
f(a+h)  f(a)  =  lim h0 
h 
^{.} h  =  f'(a)^{.} 0 = 0.  Limit of product = product of limits 
lim h0 
f(a+h)  =  lim h0 
[f(a+h)  f(a)] + f(a)  =  0 + f(a) = f(a).  Limit of sum = sum of limits 
lim xa0 
f(x)  =  f(a). 
lim xa 
f(x)  =  f(a), 