# Thread: prove function is continuous

1. ## prove function is continuous

If $f(x) = 5x - 6$, prove that f is continuous in its domain.

So,

Pick any > 0. Take any sequence { xn } converging to c. Then there exists an integer N such that
| xn - c | < /5

then,
|f(xn)-(5 c - 6)| = |5xn-6-5c+6| = 5|xn-c|<

My question is, why do we have |xn - c | < /5? Why not |xn - c | < ? Where did the 5 come from?

Thanks.

2. Originally Posted by sfspitfire23
If $f(x) = 5x - 6$, prove that f is continuous in its domain.

So,

Pick any > 0. Take any sequence { xn } converging to c. Then there exists an integer N such that
| xn - c | < /5

then,
|f(xn)-(5 c - 6)| = |5xn-6-5c+6| = 5|xn-c|<

My question is, why do we have |xn - c | < /5? Why not |xn - c | < ? Where did the 5 come from?

Thanks.
To counteract the coefficient of the x term?

3. I guess my question is how do you choose delta in continuous proofs when you dont know what the actual limit is?

See, the definition of continuity. the delta basically depends on the choice of epsilon. Moreover this function is uniformly continuous.

you have f(x)=5x+6. AS per your proof we have $x_{n} \to c \ \Rightarrow f(x_n) \to f(c)$. Here u have the delta value as epsilon/5.

5. Originally Posted by sfspitfire23
I guess my question is how do you choose delta in continuous proofs when you dont know what the actual limit is?
You don't in theory. But, I think you know what the limit should be.

It's like induction. You can't use it until you know the answer, but usually the answer is a) apparent or b) unseeable.

6. So....really.....I should perform the |f(xn)-(5 c - 6)| = |5xn-6-5c+6| = 5|xn-c|< calculation first and then see what i need to make |xn-c| less than to satisfy the |f(xn)-f(c)| calculation?

This seems odd to me.

7. Originally Posted by sfspitfire23
So....really.....I should perform the |f(xn)-(5 c - 6)| = |5xn-6-5c+6| = 5|xn-c|< calculation first and then see what i need to make |xn-c| less than to satisfy the |f(xn)-f(c)| calculation?

This seems odd to me.
Yes.

you have f(x)=5x+6. AS per your proof we have $x_{n} \to c \ \Rightarrow f(x_n) \to f(c)$. Here u have the delta value as epsilon/5.