Example a function that is continuous at every point but not derivable

In summary, the Weierstrass function and the Dirichlet function are both examples of functions that are continuous at every point but not differentiable. The Weierstrass function is a specific example of a continuous function that has a slope that changes erratically, while the Dirichlet function is discontinuous everywhere. Other examples include f(x) = sin(\frac{\pi}{x}), f(x) = |x|, and f(x) = \sqrt[3]{x} at x=0. These functions have intervals where the derivative is undefined or divides by zero, making them not differentiable on those intervals.
  • #1
hadi amiri 4
98
1
can you example a function that is continuous at every point but not derivable
 
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  • #2


The slope erratically changes.
 
  • #4


Yes, I also think of the weierstrass function is a perfect example of that.
 
  • #5


I was thinking the dirichlet function, but that's the one that's discontinuous everywhere.
 
  • #6


[tex]f(x)= sin (\fraction\pi/x)[/tex]
[tex]f(x) = |x|[/tex]
The problem's ambiguity at x=0.

Any interval on a curve where the derivative would divide by zero. [tex]f(x) = \sqrt[3]{x}[/tex] would do this at x=0.

Edit* I'm sorry if you were looking for functions that are not differentiable on any interval but are continuous.
 
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FAQ: Example a function that is continuous at every point but not derivable

What does it mean for a function to be continuous?

Continuity of a function means that the function has no sudden jumps or breaks in its graph. This means that the function can be drawn without lifting the pen from the paper.

How is continuity different from differentiability?

Continuity and differentiability are related concepts, but they are not the same. A function is continuous if it has a continuous graph, but a function is differentiable if it has a well-defined derivative at every point. A function can be continuous but not differentiable, which means it has a continuous graph but a sharp corner or break at certain points.

Can a function be continuous at every point but not derivable?

Yes, a function can be continuous but not derivable. This means that the function has a continuous graph, but it has a sharp corner or break at certain points where the derivative does not exist. An example of such a function is the absolute value function at x=0.

What is an example of a function that is continuous at every point but not derivable?

An example of a function that is continuous at every point but not derivable is the absolute value function at x=0. Another example is the function f(x) = |x|^(1/3) at x=0.

Why is it important to study functions that are continuous but not derivable?

Studying functions that are continuous but not derivable allows us to better understand the behavior of functions and their graphs. It also helps us to identify and analyze points of discontinuity and points where the derivative does not exist. This knowledge is essential in many areas of mathematics and science, including calculus, physics, and engineering.

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