- #1

thethagent

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**TL;DR Summary:**Continuity of a function, Calculus newbie, delta, epsilon,

Greetings! I have just started studying Calculus for my engineering course, and I am already facing some problems to understand the fundamental ideas regarding the continuity of a function. I'd be very much grateful if you guys spared a minute or two to help me with this question:

f(x) = 2x+1, p = 1, prove with delta and epsilon notation that it is continuous for p

How I'd start it

1 - 𝛿 < x < 1 + 𝛿 => 3 - ε < f(x) < 3+ ε

No problems so far. I'd change f(x) for its function and then I'd have a inequality for x on both sides

1 - 𝛿 < x < 1 + 𝛿 => 1 - ε/2 < x < 1 + ε/2

From this point I'm starting to have some problems. The author of the book assumes that 1 - 𝛿 = 1 - ε/2 and 1 + 𝛿 = 1 + ε/2, but I cannot fathom why it is true. For instance, I could say that x = 3, then

0 < 3 < 6 and 2 < 3 < 9

My point with it is: x shouldn't necessarily be limited by the same two things, which is why 1 - 𝛿 = 1 - ε/2 and 1 + 𝛿 = 1 + ε/2 isn't necessarily true. Yet, the book states it, so I must be missing something here. Can you help me with it?

I appreciate anyone who has read so far; thank you so much!