in my book it says that when a function f is smooth, it also means that it is bounded. I understand that a smooth function has contineous derivatives of all orders, but how can we know that the function is bounded only by knowing it is smooth?
[Math] Are all smooth functions bounded
functionsgeneral-topologyreal-analysis
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Best Answer
Not all smooth functions are bounded. For example, $f(x)=e^x$ is as smooth as they come, but is not bounded.
Even if you are looking at functions on a bounded interval, $\frac 1x$ is smooth, but unbounded on $(0,1)$.
You can produce certain restrictions for which $f$ will be bounded, however. For example, any continuous function on a compact set (for example, any continuous function on $[0,1]$) will be bounded.
Also, I highly doubt that your book says that if a function is smooth, it also means it is bounded. It sounds like you are misreading a sentence in the book to mean something it does not. Can you quote the book directly, and also tell us what book you are referring to?