A function f is said to have a removable discontinuity at...

A function $f$ is said to have a removable discontinuity at $x=c$ if lim $_{x \rightarrow c} f(x)$ exists but $f$ is not continuous at $x=c$, either because $f$ is not defined at $c$ or because the definition for $f(c)$ differs from the value of the limit. This terminology will be needed in these exercises. (a) The terminology removable discontinuity is appropriate because a removable discontinuity of a function $f$ at $x=c$ can be "removed" by redefining the value of $f$ appropriately at $x=c .$ What value for $f(c)$ removes the discontinuity? (b) Show that the following functions have removable dis- continuities at $x=1,$ and sketch their graphs. $$ f(x)=\frac{x^{2}-1}{x-1} \quad \text { and } \quad g(x)=\left\{\begin{array}{ll}{1,} & {x>1} \\ {0,} & {x=1} \\ {1,} & {x<1}\end{array}\right. $$ (c) What values should be assigned to $f(1)$ and $g(1)$ to remove the discontinuities?

A function $f$ is said to have a removable discontinuity at $x=c$ if lim $_{x \rightarrow c} f(x)$ exists but $f$ is not continuous at $x=c$, either because $f$ is not defined at $c$ or because the definition for $f(c)$ differs from the value of the limit. This terminology will be needed in these exercises.
(a) The terminology removable discontinuity is appropriate because a removable discontinuity of a function $f$ at $x=c$ can be "removed" by redefining the value of $f$ appropriately at $x=c .$ What value for $f(c)$ removes the discontinuity?
(b) Show that the following functions have removable dis- continuities at $x=1,$ and sketch their graphs.
$$
f(x)=\frac{x^{2}-1}{x-1} \quad \text { and } \quad g(x)=\left\{\begin{array}{ll}{1,} & {x>1} \\ {0,} & {x=1} \\ {1,} & {x<1}\end{array}\right.
$$
(c) What values should be assigned to $f(1)$ and $g(1)$ to remove the discontinuities?

Key Concepts

Removable Discontinuity

A removable discontinuity occurs at a point where the limit of the function exists but does not equal the function's value at that point (either because the function is not defined there or its value is different). This type of discontinuity is considered 'removable' because the function can be redefined at that point so that the limit and the function's value match, thereby restoring continuity.

Redefining a Function to Remove a Discontinuity

Redefining a function to remove a removable discontinuity involves assigning the function value at the point of discontinuity to be equal to the limit as x approaches that point. This process does not alter the behavior of the function away from the point but ensures that the function becomes continuous at that point, effectively 'repairing' the discontinuity.

Limit of a Function

The limit of a function as x approaches a given point is a fundamental concept in calculus. It describes the value that f(x) gets arbitrarily close to as x nears the point in question, regardless of the function's behavior exactly at that point. The existence of such a limit is essential for discussing continuity and different types of discontinuities.

Continuity

Continuity of a function at a point means that the function is defined at that point, the limit as x approaches that point exists, and the limit equals the function’s value at the point. This concept is central in calculus because continuous functions have many important properties, including the ability to be approximated by polynomials, and it forms the basis for many integration and differentiation techniques.

Transcript

00:01 Okay, this question just tells us a very interesting fact in calculus, which is called the removable discontinuity.

00:09 Okay, what is a removable discontinuity? that means at this point, i mean, at specific point c, we know the limit of our f exists, let's say it is equal to l, but l is not equal to the exact value, i mean, of fc.

00:35 So we don't need it to be just equal to the limit if l is not equal to the exact value of exact value, i mean, f of c.

00:47 So we see x is equal to c is a removable discontinuity for f.

01:06 So by its definition, you can see how to remove this discontinuity for a removable discontinuity.

01:19 Let's say suppose x is equal to say is a removable discontinuity for f.

01:34 How to remove it? it's very simple.

01:55 We just need to redefine fc so that fc, f of c is just equal to limit, which is l, right? what do i mean? a removable discontinuity is just like its name.

02:20 Okay, here, suppose this is c, and we know this is fc, this is l.

02:28 We know the limit of f around this point is l, while fc is not equal to l.

02:46 So we can see this point is called a removable discontinuity of our function.

02:52 How to remove this discontinuity? we just need to redefine f, i mean, we just need to move this point so that it just matches our graph, okay? this is how to remove this discontinuity of our function.

03:18 Okay, now let's consider those two examples given in this question.

03:23 The first one is x to the power 2 minus 1 divided by x minus 1.

03:32 We know this function is continuous except for x is equal to 1.

03:49 Because at this point, we don't have a definition for our f as its denominator is just equal to 0.

03:57 So now let's just consider the limit around 1.

04:03 This will be just equal to limit as x approaches 1, x plus 1 times x minus 1 divided by x minus 1.

04:15 We just factorize the numerator for our function.

04:20 And kill all the common factors.

04:24 You know, it is the limit of this one...