Differentiability and continuity

In summary, the conversation discusses the definition of a function and determining for which values it is differentiable at a given point. The test for this involves checking the continuity of both the function and its derivative, and the question is raised as to why this is necessary for differentiability. It is known that continuity is not sufficient for differentiability, but it is necessary. The method for checking continuity involves comparing two limits and ensuring they are equal.
  • #1
Yankel
395
0
Dear all,

The function f(x) is defined below:

\[\left \{ \begin{matrix} 3x^{2} &x\leq 1 \\ ax+b & x>1 \end{matrix} \right.\]

I want to find for which values of a and b the function is differential at x = 1.

The test I was given, is to check the continuity of both f(x) and f'(x). This is fairly easy technically. Checking continuity is only calculating two limits and comparing them.

My question is why this is true. Why the continuity of both f(x) and f'(x) at a point means the function is differential there. I mean, it is known that continuity does not imply differentiability...

Thank you !
 
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  • #2
Yankel said:
Dear all,

The function f(x) is defined below:

\[\left \{ \begin{matrix} 3x^{2} &x\leq 1 \\ ax+b & x>1 \end{matrix} \right.\]

I want to find for which values of a and b the function is differential at x = 1.
First the word is "differentiable", not "differential (which is a noun, not an adjective).
The test I was given, is to check the continuity of both f(x) and f'(x). This is fairly easy technically. Checking continuity is only calculating two limits and comparing them.

My question is why this is true. Why the continuity of both f(x) and f'(x) at a point means the function is differential there. I mean, it is known that continuity does not imply differentiability...

Thank you !
Yes, "continuity" is not "sufficient" for "differentiability" but it is "necessary". That is, if a function is not continuous it cannot be differentiable. Further, you are not really checking continuity of f. The derivative is not necessarily continuous but it must satisfy the "intermediate value property" so the "limit from the right" must equal the
"limit from the left". That is what you are checking.
 

1. What is the difference between differentiability and continuity?

Differentiability and continuity are related concepts in calculus, but they are not the same. Continuity refers to a function's ability to be drawn without any breaks or holes. In other words, the graph of a continuous function can be drawn without lifting the pen from the paper. On the other hand, differentiability refers to a function's ability to have a derivative at a specific point. A function is differentiable at a point if it has a well-defined tangent line at that point.

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

A function is differentiable at a point if it has a well-defined tangent line at that point. This means that the function must be continuous at that point, and the limit of the difference quotient (the slope of the secant line between two points on the function) must exist as the two points get closer and closer together. If these conditions are met, then the function is differentiable at that point.

3. Can a function be continuous but not differentiable?

Yes, it is possible for a function to be continuous but not differentiable. This can occur when the function has a sharp corner or a cusp at a specific point. In these cases, the function is continuous, but the limit of the difference quotient does not exist, so the function is not differentiable at that point.

4. What is the relationship between continuity and differentiability?

Continuity is a necessary condition for differentiability. In other words, if a function is differentiable at a point, it must also be continuous at that point. However, the reverse is not always true. A function can be continuous but not differentiable at a point, as mentioned in the previous question.

5. How do I use the concept of differentiability and continuity in real-world applications?

Differentiability and continuity are important concepts in calculus that have many real-world applications. For example, in physics, the velocity and acceleration of an object can be described using derivatives, which require the function to be differentiable. In economics, the concept of marginal cost and marginal revenue also involve derivatives. In engineering, the slope of a curve can be used to determine the rate of change of a system. In summary, differentiability and continuity are crucial for understanding and analyzing various real-world phenomena.

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