Derivatives

Differentiability and Continuity

Common reasons a function fails to be differentiable at a point include the following. These are examples rather than an exhaustive classification.

Corner

\displaystyle y=|x|

Sudden Turn

\displaystyle y=\frac23x,\quad0\le x\le\frac32

\displaystyle y=6x-8,\quad\frac32<x\le2

The slope changes abruptly where the two line segments meet.

Asymptote

\displaystyle y=\frac1{x+1}

Unequal One-sided Limits

\displaystyle y=-0.7(x+1)+2,\quad x<-1

\displaystyle y=x^2-2,\quad x\ge-1

In these cases, the difference quotient has no finite two-sided limit at the indicated point, or the function itself is undefined there. Functions are not differentiable at such points.

Differentiability implies continuity, but the reverse is not true. Consider $y=|x|$ as a counterexample. It is continuous for every real $x$ , but it is not differentiable at $x=0$ because there is a corner there. The one-sided slopes disagree.

\displaystyle y=|x|

Target input0Closeness · power of ten2

\displaystyle h=0.01,\quad m_-= -1,\quad m_+=1

At zero, the left slope is −1 and the right slope is 1. The function is continuous, but no derivative exists.

QUESTION

Which statement about

\displaystyle f(x)=|x|

\displaystyle x=0

is correct?

Discuss continuity and differentiability at each marked input

\displaystyle y=0.7x,\quad x\le-3

\displaystyle y=e^x+1,\quad-3<x\le1

\displaystyle y=(x-3)^2+e-3,\quad x>1

(a) $\displaystyle x=2$

(b) $\displaystyle x=1$

◀ Previous SectionThe First Principles Next Section ▶Basic Derivative Rules

Loading lesson… / 正在打开教程…

Leafletv4.3.3

Courses Pro Get Pro Login / Register

Derivatives

Differentiability and Continuity

Common reasons a function fails to be differentiable at a point include the following. These are examples rather than an exhaustive classification.

Corner

\displaystyle y=|x|

Sudden Turn

\displaystyle y=\frac23x,\quad0\le x\le\frac32

\displaystyle y=6x-8,\quad\frac32<x\le2

The slope changes abruptly where the two line segments meet.

Asymptote

\displaystyle y=\frac1{x+1}

Unequal One-sided Limits

\displaystyle y=-0.7(x+1)+2,\quad x<-1

\displaystyle y=x^2-2,\quad x\ge-1

In these cases, the difference quotient has no finite two-sided limit at the indicated point, or the function itself is undefined there. Functions are not differentiable at such points.

\displaystyle y=|x|

Target input0Closeness · power of ten2

\displaystyle h=0.01,\quad m_-= -1,\quad m_+=1

At zero, the left slope is −1 and the right slope is 1. The function is continuous, but no derivative exists.

QUESTION

Which statement about

\displaystyle f(x)=|x|

\displaystyle x=0

is correct?

Discuss continuity and differentiability at each marked input

\displaystyle y=0.7x,\quad x\le-3

\displaystyle y=e^x+1,\quad-3<x\le1

\displaystyle y=(x-3)^2+e-3,\quad x>1

(a) $\displaystyle x=2$

(b) $\displaystyle x=1$

◀ Previous SectionThe First Principles Next Section ▶Basic Derivative Rules

Differentiability and Continuity