Derivatives, Properties, and Formulas
Read this section to understand the properties of derivatives. Work through practice problems 111.
Theorem 
If 
a function is differentiable at a point, 

then 
it is continuous at that point. 
The contrapositive form of this theorem tells about some functions which do not have derivatives:
Contrapositive Form of the Theorem: 


If  is not continuous at a point,  

then 
Proof of the Theorem: We assume that the hypothesis, is differentiable at the point , is true so
exists and equals . We want to show that must necessarily be continuous at
Since can be written as , we have
It is important to clearly understand what is meant by this theorem and what is not meant: If the function is differentiable at a point, then the function is automatically continuous at that point. If the function is continuous at a point, then the function may or may not have a derivative at that point.
If the function is not continuous at a point, then the function is not differentiable at that point.
Example 1: Show that is not continuous and not differentiable at 2 (Fig. 1).
Solution: The onesided limits, and , have different values so does not exist, and is not continuous at 2. Since is not continuous at 2, it is not differentiable there.
Lack of continuity is enough to imply lack of differentiability, but the next two examples show that continuity is not enough to guarantee differentiability.
Example 2: Show that is continuous but not differentiable at (Fig. 2)
Solution: so is continuous at , but we showed in Section 2.1 that the absolute value function was not differentiable at .
A function is not differentiable at a cusp or a "corner".
Example 3: Show that is continuous but not differentiable at (Fig. 3)
Solution: so , and is continuous at . which is undefined at so is not differentiable at .
A function is not differentiable where its tangent line is vertical.
Practice 1: At which integer values of is the graph of f in Fig. 4 continuous? differentiable?
Graphically, a function is continuous if and only if its graph is connected and does not have any holes or breaks.
Graphically, a function is differentiable if and only if it is continuous and its graph is smooth with no corners or vertical tangent lines.