# Determine whether the function f(x) is continuous

• naspek
In summary, The function f(x) is continuous at x = 1 if and only if the limit as x approaches 1 from the left and right are equal, and the limit exists at x = 1, which is equal to f(1). In this case, the limit as x approaches 1 from the left is 0 and the limit as x approaches 1 from the right is 2, thus the function is not continuous at x = 1.
naspek

## Homework Statement

Given that

f(x) = { x + 1 ......; if x < 1
...{ 2 .....; if x = 1
...{ [4(x-1)] / (x^2 - 1) ; if x > 1

Determine whether the function f(x) is continuous at x = 1

i don't know how to start..
can someone give me an idea to start..

Try finding the limit of f as x approaches 1 from the left and the right (that is, with values of x < 1 and values of x > 1)

ok.. for

$$\lim_{x \to 1^{-}} f(x)$$

i got 0

for

$$\lim_{x \to 1^{+}} f(x)$$

i got 2

am i got it right?
so.. how can i conclude it?

ok so now you have to go back to the calc one definition of continuity. the requirements were
If f is continuous at a then, these 3 facts have to hold
$$\lim_{x \to a^{-}} f(x) = f(a)$$
$$\lim_{x \to a^{+}} f(x) = f(a)$$
$$\lim_{x \to a} f(x) = f(a)$$

naspek said:
ok.. for

$$\lim_{x \to 1^{-}} f(x)$$

i got 0
For x< 0, f(x)= 1+ x. Are you saying that $\lim_{x\to 0} 1+ x= 0$?

for

$$\lim_{x \to 1^{+}} f(x)$$

i got 2

am i got it right?
so.. how can i conclude it?
This function is continuous at x=0 only if the limit there exists and is equal to f(0). The limit itself exist only if those two one sided limits are the same.

## 1. What is continuity?

Continuity is a property of a function where there are no breaks or abrupt changes in the graph. This means that the values of the function change smoothly and continuously as the input value changes.

## 2. How can you determine if a function is continuous?

A function is continuous if it meets three criteria: 1) it is defined at every point in its domain, 2) the limit of the function at each point is equal to the actual value of the function at that point, and 3) the graph of the function has no breaks or jumps.

## 3. What is the importance of continuity in mathematics?

Continuity is an important concept in mathematics because it helps us understand and analyze the behavior of functions. It allows us to make predictions and draw conclusions about a function based on its behavior in a certain interval.

## 4. Can a function be continuous at one point but not at another?

Yes, a function can be continuous at one point but not at another. This means that the function meets the criteria for continuity at one point, but not at another. In other words, the function may have a break or jump at a certain point, but is smooth and continuous everywhere else.

## 5. How does the concept of continuity relate to differentiability?

If a function is continuous at a point, then it is also differentiable at that point. This means that the function has a well-defined derivative at that point. However, a function can be continuous at a point but not differentiable, if the derivative does not exist at that point.

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