I came across the statement
"a function is integrable provided it is continous"
and was asked to rewrite it in the form "If $P$ then $Q$".
I identified from the initial statment that the continous condition of a function is a neccassary one for a function to be integrable, but this is wrong, why is this so?
My answer was "If a function is integrable then it is continous" but this is wrong in the book it says "If a function is continous then that function is integrable"
To place emphasis, it's
A function is integrable provided it is continous.
In other words, if I provide you with a continuous function, then, according to the statement, you can deduce that that function is integrable.
Thus, given a function, $P$ is "it is continuous" and $Q$ is "it is integrable", using your notation.
Provided : make available for use , enable/allow.
In mathematics, to "provide" an assumption is to say that it is to be assumed true.Therefore, in the above, you can see that "provided the function is continuous" is asking us to assume that the function is continuous. The conclusion is then that the function is integrable.
In other words, $P \implies Q$ is the same as "if $P$ then $Q$" which is the same as "$Q$ is true if $P$ is true" (often just stated as "$Q$ if $P$"). "Provided ____" just means "if _____ is true"(or just, "if _____"). So, $P \implies Q$ is the same as "$Q$ is true provided $P$ is true".
Therefore, "a function is integrable provided it is continuous" would translate into the implication $P \implies Q$, where $P$ is "the function is continuous" and $Q$ is "the function is integrable".
Saying "a function is integrable provided it is continuous" is equivalent to saying "a function is integrable if it is continuous." From there, you just need to switch the order of the statements to end up with "If a function is continuous, then it is integrable."
