# Does $f(0) = 0 \implies f'(x) = 0$?

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Let $f(x)=x$ then $f(0)=0$ but $f'(x)=1$ so not true for all cases.

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### Anmol Bhullar

Wannabe mathematician. Currently: Undergraduate in Math/CS.

Updated on August 01, 2022

• Anmol Bhullar 3 months

Assuming $f(x)$ is differentiable $\forall x$

In my textbook, for one of the questions, it says

$f(0) = f'(0) = 0$, I was a little confused since I

thought $f(0) = 0 \implies f'(x) = 0$ and thought it was

redundant, but I'm probably wrong. If I am, can someone

perhaps give a counterexample?

• Theodoros Mpalis over 6 years
indeed. Because 0 is differentiable.
• Ian Miller over 6 years
Does $f(x)=0 \forall x$ or are you meaning at a specific point $x$?
• Anmol Bhullar over 6 years
@IanMiller sorry I made a typo I meant to say $f(0) = 0$ rather than $f(x) = 0$
• Ian Miller over 6 years
Whats the question in the book where it says $f(0)=f'(0)=0$. Some context will help us answer your problem.
• Thomas Andrews over 6 years
Okay, with the new edit, no, it is possible for $f(0)=0$ but $f'(0)\neq 0$.
• Archis Welankar over 6 years
According to your edit yes as 0 is constant
• Theodoros Mpalis over 6 years
I think he mean that $f$ is constant.
• Archis Welankar over 6 years
Yes so i wrote my comment