is continuous at a
point if three things occur:Calculus
Lesson
05
Continuity, One-sided Limits, Intermediate Value Theorem
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Back to Dr. Nandor's Calculus Page
is continuous at a
point if three things occur:
f(x) has a limit=1/5 at x=-1 and it has a function value of 2 at x=-1, so by rule 3,
it is not continuous at x=-1. f(x) has no function value at x=-6, so by rule 2, it is not
continuous at x=-6. f(x) has a function value of 1/8 at x=2, and the limit as x->-2 is
also 1/8, so f(x) is continuous at x=2.
Graphically, here are some common discontinuities:
 |
Point Discontinuity
 |
Point Discontinuity
 |
Jump Discontinuity (or filled holes)
 |
Infinite Discontinuity
Intermediate Value Thm:
if f(x) is continuous on 
and
, then there must exist some
such that 
.
Give example: there must be 
for 
and 
One-Sided Limits
Just give an example - it's like half a limit, which we've already done a lot of.
More Definitions:
Continuous AT A POINT is described above.
Continuous OVER AN INTERVAL means that the function is continuous at every
point in that interval.
A continuous FUNCTION means a function that is continuous ON EVERY POINT
IN ITS DOMAIN. Note that this means that the first point discontinuity above belongs
to a continuous function!