Practice question · Sort into groups

Sort each function by what happens at the marked point.

Groups: Continuous there · Removable hole · Jump · Vertical asymptote

$1/x$ at $x = 0$
Postage cost as weight crosses a price step
$|x|$ at $x = 0$
$\frac{x^{2}-4}{x-2}$ at $x = 2$

Hints

Try to cancel first: $(x^{2}-4)/(x-2)$ simplifies everywhere except one missing point.
$|x|$ has a sharp corner at 0, but sharp is not broken. Trace it without lifting your pen.

Show the answer

Continuous there: $|x|$ at $x = 0$

Removable hole: $\frac{x^{2}-4}{x-2}$ at $x = 2$

Jump: Postage cost as weight crosses a price step

Vertical asymptote: $1/x$ at $x = 0$

Why

The three ways continuity fails: a hole (limit exists, point missing), a jump (one-sided limits disagree), a blow-up (no finite limit). The corner of $|x|$ breaks none of these, corners are a problem for derivatives, not for continuity.

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Sort each function by what happens at the marked point.

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