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Continuity at a Point
If the limit does not exist, or if it exists but does not equal then the function is discontinuous at c
Jump Discontinuity
• If the one side limits exist but are not equal. This is considered bad because it is not an easy fix.
𝐹 (𝑥 )={ 𝑥 𝑓𝑜𝑟 𝑥<13 𝑓𝑜𝑟 1≤ 𝑥≤3𝑥 𝑓𝑜𝑟 𝑥>3
• Continuous at all points except at 1.
• Jump discontinuity at 1
• Right-continuous at 1
Infinite Discontinuity
• If one or both side limits of a function are infinite, then the function is infinite discontinuous at that point.
Basic Laws of Continuity
• If and are continuous at , the following functions are also continuous at
for any constant
if
Continuity of Polynomial and Rational Functions
• Let and be polynomials. Then:
is continuous on the real line.is continuous on its domain (at all values such
that )
Continuity of Some Basic Functions
• is continuous on its domain for n a natural number.
• and are continuous on the real line.• is continuous on the real line (for ). • is continuous for (for ).
Continuity of Composite Functions
• If is continuous at and is continuous at then the composite function is continuous at