According to Wikipedia, a smooth function is a function that has derivatives of all orders. I don't understand what this means if the case was for example the function $$f(x) = 1+2x$$ This can be differentiated only twice until it is zero. Is this considered a smooth function? I am confused about the definition.
[Math] a smooth function
definitionderivativesreal-analysisterminology
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Best Answer
The function $h(x)=0$ is differentiable with $h'(x)=0$. So yes your function is smooth.
Note that in some contexts smooth can mean different things, such as 1 or 2 times differentiable. Sometimes we don't need all derivatives, so smooth just means "as many as I need".