calculus question?

Question

prove that if lim f(c + delta x) = f(c) as delta x approaches 0, then f is continuous at c

Answer 1

Suppose [Δx→0]lim f(c+Δx) = f(c). Then by the definition of limit:

∀ε>0, ∃δ>0 s.t. 0 < |Δx - 0| < δ ⇒ |f(c+Δx) - f(c)| < ε

Now, let ε>0 be arbitrary. Choose δ so as to satisfy the definition of limit given above. Then suppose |x-c| < δ. Then either x=c or x≠c. If x=c, then |f(x) - f(c)| = |f(c) - f(c)| = 0 < ε. Conversely, if x≠c, then we have that |x-c| ≠ 0. So 0 < |(x-c) - 0| < δ, and so since δ was chosen to satisfy the limit definition, we have that |f(c + (x-c)) - f(c)| < ε. Of course, |f(c + (x-c)) - f(c)| = |f(x) - f(c)| < ε. Thus in either case we have |x-c| < δ ⇒ |f(x) - f(c)| < ε. Since we can choose such a δ for any ε>0, we have shown:

∀ε>0, ∃δ>0 s.t. |x-c|<δ ⇒ |f(x) - f(c)|<ε

Which is to say that f is continuous at c. Q.E.D.

Answer 2

This is very complicated to just prove with text. It requires graphs and a lot of other things. I forgot exactly what the proof was but it's like "limits boundary proof" where you draw boundaries proving that for any delta x and any delta y, it'll never pass through the rectangle it's drawn into.