True or false statement about a $C^\infty$ function If $f \in  C^\infty$ and $f^k(0)=0 \forall$  $k \in \mathbb N \bigcup \{0\}$,then can we conclude that $f$ is identically zero?
What does mean $\mathbb{C}^\infty$? If you wanted to say just $C^\infty$, then the answer to your question is negative as Matthew Leingang showed. Another classical example is the following: Consider the function $f(x)=\exp(-\frac{1}{x})$ if $x&gt;0$, and $f(x)=0$ otherwise. It is not so difficult to show that $f$ is a $C^\infty$ function on the real line. Moreover, this kind of functions are very interesting because they play an important role in the theory of $C^\infty$ functions. As you easily can note, the examples give them here are functions that are not analytic, they just are smooth functions.
What does mean $\mathbb{C}^\infty$? If you wanted to say just $C^\infty$, then the answer to your question is negative as Matthew Leingang showed. Another classical example is the following: Consider the function $f(x)=\exp(-\frac{1}{x})$ if $x&gt;0$, and $f(x)=0$ otherwise. It is not so difficult to show that $f$ is a $C^\infty$ function on the real line. Moreover, this kind of functions are very interesting because they play an important role in the theory of $C^\infty$ functions. As you easily can note, the examples give them here are functions that are not analytic, they just are smooth functions.