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I'm having some trouble to solve the following exercise:

Supposing that \(\displaystyle |f(x) - f(1)|≤ (x - 1)^2\) for every \(\displaystyle x \).

Show that \(\displaystyle f\) is continuous at \(\displaystyle 1\)

(Sorry if the text seems a bit weird, but it's because I'm still getting used to translate all these math-related terms to english.)

I know that if f is continuous at 1, the following will be truth:

\(\displaystyle

0<|x-1|< \delta\) \(\displaystyle ⇒\) \(\displaystyle |f(x) - f(1)| < \epsilon\)

I thought of choosing \(\displaystyle \delta = \epsilon/2(x-1)^2\), then I would find that \(\displaystyle |f(x) - f(1)| < \epsilon/2 < \epsilon\)

But, as far as I know, choosing a \(\displaystyle \delta\) that depends on \(\displaystyle x\) is wrong.

I really don't know what to do.

Thank you,

Bueno.