he definition of the limit of a function is as follows: Let be a function defined on a subset [PLAIN]https://upload.wikimedia.org/math/a/1/b/a1b67abab803e714098f3e69a33900da.png, [Broken] let be a limit point of [PLAIN]https://upload.wikimedia.org/math/f/6/2/f623e75af30e62bbd73d6df5b50bb7b5.png, [Broken] and let be a real number. Then the function has a limit at is defined to mean for all [PLAIN]https://upload.wikimedia.org/math/b/0/f/b0f19c5714fe9f9891ed26ff783cf639.png, [Broken] there exists a https://upload.wikimedia.org/math/1/c/b/1cb24dafc8035d2c720256620066ae73.png such that for all in that satisfy [PLAIN]https://upload.wikimedia.org/math/6/e/e/6ee6dc9ee03f04da6ff784b6eca416e5.png, [Broken] the inequality https://upload.wikimedia.org/math/2/c/4/2c412a6f49514db4dbd890537413bcc6.png holds. . Taken from https://en.wikipedia.org/wiki/(ε,_δ)-definition_of_limit The problem what Iam facing is the statement ," for all in that satisfy [PLAIN]https://upload.wikimedia.org/math/6/e/e/6ee6dc9ee03f04da6ff784b6eca416e5.png". [Broken] Though I have understood this statement, this hinders my proving this property of limit lim f(c+h)=L =lim f(x). h->0 x->c I am trying to prove the LHS from the RHS. We know, "the function f has a limit L at is defined to mean for every ε>0,there exists a δ>0, sucht that, for all x in D satisfying, o<|x-a|<δ, implies |f(x)-L|<ε. The LHS can be proven from the trivial fact taking x-c=h. A question arises to me. How do we know the existance of such an h? Also, for what all h does this inequality o<|h|<δ implies |f(c+h)-L|<ε hold?