Lesson 2.6: Differentiability: A function is differentiable at a point if it has a derivative there.
Example 1:
If f(x) is differentiable at x = a, then f(x) is also continuous at x = a.
f(x) − f(a)
(f(x) − f(a)) = lim.
(x − a) · f(x) − f(a) x − a This is okay because x − a = 0 for limit at a.
(x − a) lim.
f(x) − f(a)
Continuous:
A function is said to be continuous on the interval [a,b] if it is continuous at each point in the interval. Note that this definition is also implicitly assuming that both f(a) and limx→af(x) lim x → a exist. If either of these do not exist the function will not be continuous at x=a
Answers & Comments
Answer:
Differentiable:
Lesson 2.6: Differentiability: A function is differentiable at a point if it has a derivative there.
Example 1:
If f(x) is differentiable at x = a, then f(x) is also continuous at x = a.
f(x) − f(a)
(f(x) − f(a)) = lim.
(x − a) · f(x) − f(a) x − a This is okay because x − a = 0 for limit at a.
(x − a) lim.
f(x) − f(a)
Continuous:
A function is said to be continuous on the interval [a,b] if it is continuous at each point in the interval. Note that this definition is also implicitly assuming that both f(a) and limx→af(x) lim x → a exist. If either of these do not exist the function will not be continuous at x=a