Question

Use the definition of continuity to show f(x)=x^2 is continuous everywhere on R

Answer 1

Maybe better i say it like this, sry...:)
for small enough increment of x you need small enough increment of f(x).
\[\Delta f(x) = f(x+ \Delta x) - f(x) = x ^{2} + 2 \Delta x + \Delta x ^{2} - x ^{2} = 2 \Delta x + \Delta x ^{2}\]
If you take limit of this:\[\lim_{\Delta x \rightarrow 0} \Delta f(x) = \lim_{\Delta x \rightarrow 0} 2 \Delta x + \Delta x ^{2} = 0\]
So the increment of function gets small, togehter with x

Answer 2

@studybird