lim
f(x) - f(c) exists at x = c
x®c
x - c
Prove: f(x) is continuous. (Use the 3-step proof)
PROOF:
i) f(c) exists because the given limit exists and f(c) must be used to calculate the limit
ii) lim [f(x) - f(c)]
= lim [f(x) - f(c)](x - c)
x®c
x®c
x - c
= lim [f(x) - f(c)] (x - c)
x®c x - c
= lim f(x) - f(c) lim (x - c)
x®c x - c
= [f ' (c)](0)
= 0
Since lim
[f(x) - f(c)] = 0, then lim f(x) =
f(c) and the limit exists
x®c
x®c
iii) lim f(x) = f(c)
x®c
Therefore, f(x) is continuous at x = c
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