Differentiability of weird function

What does that tell you about the limit?In summary, the functions f(x) and g(x) are defined differently depending on whether x is rational or irrational. Both functions are continuous at 0 and discontinuous at all other values of x. To show that f is not differentiable at 0, we can use the definition of the derivative and see that the limit is undefined because it approaches different values depending on whether x is rational or irrational. On the other hand, g is differentiable at 0 and its derivative at 0, g'(0), is equal to 0. This is because the limit of g(x)/h as h approaches 0 approaches 0 regardless of whether x is rational or irrational.
  • #1
Ara macao
27
0
Let f(x) = x if x rational and f(x) = 0 if x is irrational

Let g(x) = x^2 if x rational and g(x) = 0 if x is irrational.

Both functions are continuous at 0 and discontinuous at each x != 0.

How do I show that f is not differentiable at 0?
How should I show that g is differentiable at 0? Give g'(0) as well.

Thanks...
 
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  • #2
Use the definition: [tex]f'(0)= lim_{h->0}\frac{f(h)}{h}[/tex].
but [tex]\frac{f(h)}{h}[/tex] is 1 if x is rational, 0 if x is irrational. What does that tell you about the limit?

[tex]\frac{g(h)}{h}[/tex] is h is x is rational, 0 if x is irrational.
 

FAQ: Differentiability of weird function

1. What is the definition of differentiability of a function?

The differentiability of a function is a mathematical concept that describes the smoothness of a function. A function is considered differentiable at a certain point if it has a well-defined derivative at that point, meaning that the function is continuously changing and has a well-defined slope at that point.

2. Can a function be differentiable at some points and not at others?

Yes, a function can be differentiable at some points and not at others. This is because differentiability is dependent on the existence of a well-defined derivative at a specific point. If the function is not continuously changing or if there is a discontinuity at that point, the function will not be differentiable at that point.

3. What are some common examples of non-differentiable functions?

Some common examples of non-differentiable functions include absolute value, step functions, and functions with sharp corners or cusps. These types of functions have discontinuities or abrupt changes, making it impossible for them to have a well-defined derivative at those points.

4. How is differentiability related to continuity?

Differentiability and continuity are closely related concepts. A function is differentiable at a point if it is also continuous at that point. This means that a function must be continuous in order for it to be differentiable at a specific point. However, a function can be continuous at a point but not differentiable, as there may be a sharp change or discontinuity at that point.

5. Can a function be differentiable but not continuous?

No, a function cannot be differentiable if it is not continuous. As mentioned before, differentiability requires continuity at a certain point. If a function is not continuous at a specific point, it cannot have a well-defined derivative at that point, thus it is not differentiable.

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