Waste less time on Facebook — follow Brilliant.
×

The relationship between differentiability and continuity

Name a function that is everywhere continuous but nowhere differentiable. Or, if such a function does not exist, explain why.

Note by Hobart Pao
2 years, 1 month ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

The Weierstrass function is the most commonly cited example. Here and here are other examples, not to mention fractals.

It could be possible that "almost all" everywhere continuous functions are nowhere differentiable, but I'll have to give that one a bit more thought.

Brian Charlesworth - 2 years, 1 month ago

Log in to reply

Yup, that's what I was thinking of.

Hobart Pao - 2 years, 1 month ago

Log in to reply

If you, by the word 'everywhere', mean everywhere in its domain, then I can give you a much simpler function: Define the function \(f:N\rightarrow R\) as \(f\left( n \right)=n,\forall n\in N\). Clearly, since each point in its domain is isolated, we can say that it is continuous 'everywhere' in its domain by default, but not differentiable anywhere.

Log in to reply

Good point. By most definitions of continuity a discrete function, (i.e., a function in which its domain is at most countable), it is considered "vacuously true" that the function is continuous at all of its (isolated) points. We could also then consider the function \(f:\mathbb{Q} \rightarrow \mathbb{R}, f(x) = x\) for all \(x \in \mathbb{Q}\) as an example.

Brian Charlesworth - 2 years ago

Log in to reply

Yes yes I was trying to point that out only.

Log in to reply

Oh sorry I didn't see your post is already replied.

Ravi Dwivedi - 2 years, 1 month ago

Log in to reply

Weierstrass function is such a function

Ravi Dwivedi - 2 years, 1 month ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...