Analyzing Continuity and Differentiability of f(x) at x=1 & x=3

In summary, the conversation discusses the continuity and differentiability of a piecewise function, f(x), defined as |x-3| for x>=1 and (x^2/4)-(3x/2)+(13/4) for x<1. It is determined that the function is continuous but not differentiable at x=3 due to being a modulus function. However, there is confusion about the continuity and differentiability at x=1, as taking the limit from both sides does not result in the same value for both parts of the function. After realizing a mistake in the arithmetic, it is concluded that the function is indeed continuous at x=1.
  • #1
chaoseverlasting
1,050
3

Homework Statement


f(x) is a piecewise function defined as:

[tex]|x-3| x>=1[/tex]
[tex]\frac{x^2}{4}-\frac{3x}{2}+\frac{13}{4} x<1[/tex]

Discuss the continuity and differentiability of this funtion at x=1 and x=3

Homework Equations





The Attempt at a Solution


At x=3, this function is continuous but not differentiable, being a modulus funtion. But how is is continuous and differentiable at x=1? Putting limit x=1 in the above funtion doesn't give you the same value for both the parts!
 
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  • #2
Look at the limit as x->1 from both sides of the function and the derivative of the function. What do you get as answers to these four questions?
 
  • #3
The func is cont. at x=1. Plug in 1 again, in both expression
 
  • #4
Whoa! I really feel like an idiot now. I messed up the arithemetic.
 

Related to Analyzing Continuity and Differentiability of f(x) at x=1 & x=3

What is continuity and differentiability?

Continuity and differentiability are properties of a function that describe how smoothly it behaves. A function is continuous if there are no breaks or gaps in the graph, meaning that the function can be drawn without lifting your pencil. A function is differentiable if it has a well-defined derivative at every point, meaning that the slope of the function can be calculated at any point on the graph.

How can I determine if a function is continuous at a specific point?

To determine if a function is continuous at a specific point, you can use the continuity test which states that a function is continuous at a point if the limit of the function as x approaches that point is equal to the value of the function at that point. In other words, the left-hand and right-hand limits of the function must be equal at the specific point.

What is the difference between continuity and differentiability?

The main difference between continuity and differentiability is that continuity is concerned with the behavior of a function at a specific point, while differentiability is concerned with the behavior of a function at every point. A function can be continuous at a point but not differentiable, while a function that is differentiable at a point must also be continuous at that point.

How can I determine if a function is differentiable at a specific point?

To determine if a function is differentiable at a specific point, you can use the differentiability test which states that a function is differentiable at a point if the left-hand and right-hand derivatives at that point exist and are equal. In other words, the slope of the function must be consistent from both sides of the point.

What is the significance of analyzing continuity and differentiability of a function?

Analyzing continuity and differentiability of a function is important because it helps us understand the behavior and properties of the function. It allows us to determine if a function has any breaks or gaps, and if it has a well-defined slope at every point. This information can be used to make predictions and solve problems in various fields such as physics, engineering, and economics.

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