Check if a function is continuous at a point or over its domain
Select Function Type
Polynomial: ax^n + bx^(n-1) + ...
f(x)=x^
Rational: 1/(ax+b)
f(x)=1/(x+)
Check continuity at x=a where f changes
f(x)={x for x<,x for x>=
Test f(x)=(ax+b)/(cx+d) at x=p
f(x)=(x+)/(x+) at x=
Result
Derivation
Continuity Definition
f continuous at a if: lim_{x->a} f(x) = f(a)
1) f(a) defined 2) lim exists 3) lim = f(a)
Polynomials: continuous everywhere
Rational: continuous except denominator=0
Continuity means the graph can be drawn without lifting the pen. A function is continuous at a point if the function value, left limit, and right limit all agree. Polynomials are always continuous; rational functions have discontinuities at denominator zeros.
⚠Three conditions for continuity at a: f(a) defined, limit exists, limit = f(a). Break any one and you have a discontinuity.
What Is Continuity?
Continuity is a fundamental concept in calculus. A continuous function has no breaks, jumps, or holes. Polynomials are continuous on R. Rationals have discontinuities at points where denominator=0 (vertical asymptotes or holes if numerator also zero).
Polynomials
Always continuous on R. No breaks, holes, or asymptotes. lim x->a P(x) = P(a) for all a.
Rational Functions
Continuous except where denominator=0. Vertical asymptote if numerator != 0. Hole if both are zero.
Piecewise
Check continuity at the boundary point. Compare left limit, right limit, and function value. Must all match.
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