MATH 01130 · Limits & continuity
Continuity checklist
Definition
f is continuous at a if lim x→a f(x) exists, f(a) is defined, and lim x→a f(x) = f(a). On an interval, this must hold at every point (with one-sided limits at endpoints).
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Three failure modes: hole (limit exists, f(a) missing or wrong), jump (one-sided limits disagree), vertical asymptote (infinite).
Polynomials, rationals (on their domain), sin/cos/exp/ln(on domain) are continuous where defined.
IVT: if f continuous on [a,b] and k between f(a) and f(b), some c in (a,b) has f(c)=k — root-finding gold.
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1.Classify discontinuity of f(x)=(x²−1)/(x−1) at x=1 after defining f(1)=3 vs f(1)=2.