3.2 Continuity and Discontinuity - Introduction to Calculus

Continuity

It seems like sometimes we can evaluate a limit by just plugging in, but sometimes we can't!

Evaluate $\displaystyle \lim\limits_{x\to1}\left(\ln(\sqrt{x})+\frac1x\right)$

A function is continuous at a point $a$ if each of the following conditions holds:

$\lim\limits_{x\to a}f(x)$ exists.
$f(a)$ exists.
$\lim\limits_{x\to a}f(x)=f(a)$

Keep the cosine curve fixed for every nonzero input. Change only the function value at zero, or remove it entirely.

\displaystyle f(x)=\cos x,\quad x\ne0

Value at zero-1 Define f(0)

The limit exists and equals 1.
The function value exists: f(0) = -1.
The continuity conditions are not all satisfied.

Discontinuity

Kinds of discontinuities:

Essential

An infinite discontinuity, called “essential” in the source, occurs in $y=\frac1{x+1}$ at $x=-1$ The nearby values are unbounded.

\displaystyle y=\frac1{x+1}

Removable

A removable discontinuity is a hole: the limit exists, but the function value is missing or does not agree with it. This sine curve has its point at $x=1$ removed.

\displaystyle y=\sin x,\quad x\ne1

Jump

A jump discontinuity occurs when the one-sided limits are finite but different. This three-part function has jumps at $x=-2$ and $x=2$

\displaystyle y=\frac12x+3,\quad x<-2

\displaystyle y=0,\quad -2\le x\le2

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

So far, we know that evaluating $\lim\limits_{x\to a}f(x)$ is easy if $f(x)$ is continuous at $a$ But what if it isn't? The limit may still exist.

When we write that a limit is $\infty$ or $-\infty$ , we describe unbounded growth in a specified direction. Infinity is not a finite real limit or a value obtained by dividing by zero.

QUESTION

Both one-sided limits at

\displaystyle a

equal

\displaystyle 4

Which value makes

\displaystyle f

continuous at

\displaystyle a

Evaluate the following limits.

(a) $\displaystyle \lim\limits_{x\to3^+}\frac1{3-x}$

(b) $\displaystyle \lim\limits_{x\to3^-}\frac1{3-x}$

(d) $\displaystyle \lim\limits_{x\to-\dfrac\pi2^+}\sec(x)$

Continuity

It seems like sometimes we can evaluate a limit by just plugging in, but sometimes we can't!

Evaluate $\displaystyle \lim\limits_{x\to1}\left(\ln(\sqrt{x})+\frac1x\right)$

A function is continuous at a point $a$ if each of the following conditions holds:

$\lim\limits_{x\to a}f(x)$ exists.
$f(a)$ exists.
$\lim\limits_{x\to a}f(x)=f(a)$

Keep the cosine curve fixed for every nonzero input. Change only the function value at zero, or remove it entirely.

\displaystyle f(x)=\cos x,\quad x\ne0

Value at zero-1 Define f(0)

The limit exists and equals 1.
The function value exists: f(0) = -1.
The continuity conditions are not all satisfied.

Discontinuity

Kinds of discontinuities:

Essential

An infinite discontinuity, called “essential” in the source, occurs in $y=\frac1{x+1}$ at $x=-1$ The nearby values are unbounded.

\displaystyle y=\frac1{x+1}

Removable

A removable discontinuity is a hole: the limit exists, but the function value is missing or does not agree with it. This sine curve has its point at $x=1$ removed.

\displaystyle y=\sin x,\quad x\ne1

Jump

A jump discontinuity occurs when the one-sided limits are finite but different. This three-part function has jumps at $x=-2$ and $x=2$

\displaystyle y=\frac12x+3,\quad x<-2

\displaystyle y=0,\quad -2\le x\le2

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

So far, we know that evaluating $\lim\limits_{x\to a}f(x)$ is easy if $f(x)$ is continuous at $a$ But what if it isn't? The limit may still exist.

When we write that a limit is $\infty$ or $-\infty$ , we describe unbounded growth in a specified direction. Infinity is not a finite real limit or a value obtained by dividing by zero.

QUESTION

Both one-sided limits at

\displaystyle a

equal

\displaystyle 4

Which value makes

\displaystyle f

continuous at

\displaystyle a

Evaluate the following limits.

(a) $\displaystyle \lim\limits_{x\to3^+}\frac1{3-x}$

(b) $\displaystyle \lim\limits_{x\to3^-}\frac1{3-x}$

(d) $\displaystyle \lim\limits_{x\to-\dfrac\pi2^+}\sec(x)$

Continuity and Discontinuity

Continuity

Evaluate by continuity

Repair a removable discontinuity

Discontinuity

Essential

Removable

Jump

KNOWLEDGE QUIZ

Track unbounded one-sided behavior

Continuity and Discontinuity

Continuity

Evaluate by continuity

Repair a removable discontinuity

Discontinuity

Essential

Removable

Jump

KNOWLEDGE QUIZ

Track unbounded one-sided behavior