Realan1258

1. A function 𝑓: 𝑀 → 𝑀1 is called continuous in a point 𝑚 ∈ 𝑀 if it is defined in some neighborhood of 𝑚 and ∀𝜀 > 0 ∃𝛿(𝜀): ∀𝑥 ∈ 𝐵𝛿(𝜀) (𝑚) ⇒ 𝑓(𝑥) ∈ 𝐵𝜀 (𝑓(𝑚)). Here 𝐵𝑟 (𝑦) denotes the open ball centered at 𝑦 with the radius 𝑟 in the corresponding metric space, 𝐵𝑟 (𝑦) = {𝑥 ∈ 𝑀: 𝜌(𝑥, 𝑦) < 𝑟}. a. It is evident for 𝑐 = 0. For nonzero 𝑐 estimate |𝑐𝑓(𝑥) − 𝑐𝑓(𝑚)| = |𝑐||𝑓(𝑥) − 𝑓(𝑚)|, 𝜀 which is < 𝜀 for ∀𝑥 ∈ 𝐵𝛿(𝜀) (𝑚) (because for those 𝑥 |𝑓(𝑥) − 𝑓(𝑚)| < 𝑐). 𝑐 b. Given 𝜀 > 0 and having 𝛿𝑓 (𝜀) and 𝛿𝑔 (𝜀) from the definition of continuity, use 𝜀 𝜀 𝛿(𝜀) = min (𝛿𝑓 ( ), 𝛿𝑔 ( )) > 0 2 2 which is sufficient: |(𝑓(𝑥) + 𝑔(𝑥)) − (𝑓(𝑚) …