What is the relationship between smooth functions and functions with continuous derivative?
How does a function having continuous derivative makes it smooth? i.e no rough edges.
How can i interpret this geometrically?
Smooth functions and functions with continuous derivative
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calculus
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3See this http://math.stackexchange.com/questions/472148/smooth-functions-or-continuous – 2017-01-02
3 Answers
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By definition, a smooth function has a continuous first derivative and a continuous second, third, fourth, etc. all continuous.
So a smooth function is also a function with a continuous derivative. I'm not sure what else you'd want to get out of that.
On the other hand, a function with a continuous derivative isn't necessarily smooth. Take for example, the following function:
$$f(x)=\begin{cases}x^2&;x\ge0\\x^3&;x<0\end{cases}$$
You can differentiate it once, and it will have a continuous derivative. But if you differentiate it twice, it won't be continuous, and hence, it will not be smooth.
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A function with continuous derivative is a $C^1$-function. A function is smooth if it is infinitely derivable, that is $f'=f^{(1)}$ exists and $f^{(n+1)}=(f^{(n)})'$ exists for every integer $n>0$.
A $C^1$ function is not necessarily smooth. Consider $f(x)=|x|$ defined on $R$, you can define the primitive $g(x)=\int_0^x|t|dt$. Its derivative is $f(x)$ thus is continuous, but $f"$ is not defined since the absolute value is not derivable at $0$.
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A function is smooth over an interval if it has continuous derivatives (2 or more) up to an order over that interval. So if you are looking to see if it is smooth on the domain the yes, no rough edges are a requirement.