Calculus 2.
4 Differentiability and Continuity Notes
Write your questions
and thoughts here!
A graph of a function is shown below. Write down its equation on line #1.
y
1. 𝑦 ____________________________
2. 𝑦 ____________________________
x
3. 𝑦 ____________________________
Differentiability:
The derivative exists for each point in the domain. The graph must be a smooth line
or curve for the derivative to exist. In other words, the graph looks like a line if you zoom in.
The derivative fails to exist where the function has a
1. Discontinuity 2. Corner or cusp 3. Vertical tangent
Identify points where the function below is not continuous and/or not differentiable.
y
x
True or False True or False
Differentiability implies continuity. Continuity implies differentiability.
2.4 Differentiability and Continuity
Calculus
Practice
Identify any 𝒙-values of the function that are not continuous and/or not differentiable.
y
1.
𝑥 -values where the function is not continuous. 𝑥 -values where the function is continuous, but not
differentiable.
y
2.
𝑥 -values where the function is not continuous. 𝑥 -values where the function is continuous, but not
differentiable.
y
3.
𝑥 -values where the function is not continuous. 𝑥 -values where the function is continuous, but not
differentiable.
2.4 Differentiability and Continuity Test Prep
4. 𝑓 is continuous for 𝑎 𝑥 𝑏 but not differentiable for some 𝑐 such that 𝑎 𝑐 𝑏. Which of the following
could be true?
(A) 𝑥 𝑐 is a vertical asymptote of (B) lim 𝑓 𝑥 𝑓 𝑐 (C) The graph of 𝑓 has a cusp at
→
the graph of 𝑓. 𝑥 𝑐.
(D) 𝑓 𝑐 is undefined. (E) None of the above
5. If 𝑔 is differentiable at 𝑥 𝑐, which of the following must be true?
I. 𝑔 is continuous at 𝑥 𝑐.
II. lim 𝑔 𝑥 exists.
→
III. lim exists.
→
(A) I only (B) II only (C) III only
(D) I and II only (E) I, II, and III
6. Let ℎ be the function given by ℎ 𝑥 |𝑥 4|. Which of the following statements about ℎ are true?
I. ℎ is continuous at 𝑥 4.
II. ℎ is differentiable at 𝑥 4.
III. ℎ has an absolute minimum at 𝑥 4.
(A) I only (B) II only (C) III only
(D) I and III only (E) II and III only
7. If 𝑓 is a differentiable function such that 𝑓 2 5 and 𝑓 2 7, which of the following statements could be
false?
(A) lim 𝑓 𝑥 5 (B) lim 𝑓 𝑥 lim 𝑓 𝑥 (C) lim 7
→ → → →
(D) lim 7 (E) lim 𝑓 𝑥 7
→ →
