PLZZZZZ SOVE DIS WID XPLANATION .
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if f : R tO R is a cts funtion s.t f(x) tends to 0 as x tends to infinity show that f is unformly cts?
I disagree with the theorem. I think you could probably show that $\displaystyle e^{-x}$ is not uniformly continuous on the real line, but it satisfies all the conditions of your theorem. Now, if in addition you require that the function goes to zero as x approaches plus or minus infinity, I might believe that.
This looks to be a generalization of the Heine-Cantor theorem.
Two comments: 1. This forum is to help people get unstuck; it is not to do their thinking for them. 2. Please stop using whatever non-spelling you're using. Write using regular English. If you're studying for an M. Phil. degree, you ought to be able to spell correctly. I, for one, am having a hard time understanding you. Remember this: don't write so that people can understand you. Write so that no one can misunderstand you!
Now then. The exponential function is everywhere continuous, but it is not uniformly continuous on the interval $\displaystyle [0,\infty)$. It stands to reason, then, that the function $\displaystyle e^{-x}$ would not be uniformly continuous on the interval $\displaystyle (-\infty,0]$. Just flip everything about the y axis. That is why I believe your original theorem was incorrect. However, if you are restricting yourself only to functions on the interval $\displaystyle [0,\infty)$, your theorem would be correct. In that case, follow the links I posted in Post #4. That will give you some information, hopefully enough to start a proof.