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.

#Calculus #Differentiability

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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Comments

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 - 5 years, 8 months ago

Yup, that's what I was thinking of.

Hobart Pao - 5 years, 8 months ago

Weierstrass function is such a function

Ravi Dwivedi - 5 years, 8 months ago

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

Ravi Dwivedi - 5 years, 8 months ago

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.

Kuldeep Guha Mazumder - 5 years, 6 months ago

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 - 5 years, 6 months ago

Yes yes I was trying to point that out only.

Kuldeep Guha Mazumder - 5 years, 6 months ago

Suppose given a linear system Ax = b (1),

Renee Degutis - 2 years, 7 months ago

Math	Appears 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}`	$\boxed{123}$