Proof that f(x) = x^2 is Uniformly Continuous on (0, 1)

[@acdude5266]

Nice choice of a manageable function to illustrate the concept / definition of uniform continuity.

[@zaidbutt4332]

i cant understand the step how dell is equal to epsilon by 2

[@fellipeparreiras4435]

Cant we just take δ=ε? Because
| 𝑥 - 𝑦 | < | 𝑥 | + | - 𝑦 | = | 𝑥 | + | 𝑦 |, since 𝑥 and 𝑦 < 1, then | 𝑥 | + | 𝑦 | implies | 𝑥 - 𝑦 | < 2 which we assume is < δ.
Then this also implies, by the same manner, that | 𝑥² - 𝑦² | < 2 < δ < ε.
So, then, couldn't δ=ε?

[@mayurgo10]

Could you please explain how did you take |x+y|

[@TheMathSorcerer]

X and y are in (0,1) so they are both less than 1 so their sum is less than 2. Also The absolute value doesn't matter because they are both positive

[@mayurgo10]

thank you.@@TheMathSorcerer

[@TheMathSorcerer]

@@mayurgo10 np friend😄

[@Gaelrenaultdu06]

Hello, is it possible to prove that x^2 is continuous on R ?

[@nonono5520]

Gaël Renault yes

[@sureshkaruppasamy6257]

Sir what about 0 to infinity

[@pradeepsao3333]

Prove that set of rational numbers is
dense in the set of real numbers.

[@gauravbairwa1703]

Will it continous on [0,infinty)

[@prakashbadu4137]

Is this function is uniformly continuous in whole real line??

[@massimilianomessina5228]

no

[@prakashbadu4137]

@@massimilianomessina5228 why..?? Can you explain pls

[@massimilianomessina5228]

@@prakashbadu4137 Just look at the graph, you cant have a delta that "holds for the whole function"

[@mr_desi7102]

You are best

[@Akash-ls5pz]

tomorrow is my exam it help me a lot ....thanks sir

[@蒙娜麗莎的味道]

Hi I have a question,can u answer me plz. how to proof a function 1/x^2+sinx+2 is uniformly Conti. on real numbers?

[@thelastcipher9135]

If delta=epsilon/2 then 2delta=epsilon but we want 2delta less than epsilon so I think you made a mistake there.

[@urossavatovic3481]

TheLast Cipher no we want the absolute value to be less then epsilon witch is eaqual to 2delta

[@urossavatovic3481]

TheLast Cipher if the absolute value is less than 2delta wich is eaqual to epsilon therefore the absolute value is less then epsilon witch ends the proof