# Questions tagged [continuity]

Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) or (general-topology)

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### Proving continuity of the following piecewise function

Consider the following function : $$f(x) = \begin{cases} x^2-9 \ & \text{if}\ x \le 4 \\[2ex] \frac{2x^2-9x+4}{x-4} & \text{if}\ x >4 \end{cases}$$ Now I want to show that this function ...
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### Example of a function such that $A$ and $\alpha$ depend on $\epsilon$ from Spivak's Calculus

The following are two conditions needed for a proof below: $(i)$ if $x$ and $y$ are in $[a,b]$ and $|x-y|<\delta_{1}$, then $|f(x)-f(y)|<\epsilon$, $(ii)$ if $x$ and $y$ are in $[b,c]$ ...
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