continuity

introduction

%%visits: 4 If you find a range for the outputs, I can find a range for the inputs, that the output will always be in.

f ∈ C[a, b] := f is continuous and f(a) = lim_x → a+f(x) and $f(b) = \lim_{x \to b_{_}}f(x)$

intuition

“nearby points in the domain map to nearby points in the range” Epsilon is small. Any change in x should mean that the change in y is not too “jumpy”. continuity and [[convergence]] lim_x → x0f(x) = y₀ ⇔ ∀ sequence x_n ∈ (a, b) : lim_{n → ∞}x_n = x₀ we have lim_{n → ∞}f(x_n) = y₀

If f, and g are continuous - f + g is continuous - fg is continuous - f(g) is continuous

Continuous in general means continuous at every point. ## rigour ## exam clinic ## resources tags math__real_analysis:__complex_analysis:

backlinks

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