Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2.
1.4 Continuity
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Transcript of 1.4 Continuity
ContinuityContinuity
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3 4 5
In the 5 pictures below, let’s look at:a. Is the function defined at c ?b. Does the limit exist at c ?
a. f(c) is undb. lim DNE
a. f(c) is undb. lim EXISTS
a. f(c) is definedb. lim DNE
a. f(c) is definedb. lim EXISTS
a. f(c) is definedb. lim EXISTS
cc
c c c
So which of these are continuous?
* All three of these must occur.
A function is continuous at c if…
• 1. The function is defined at c.• 2. The limit exists at c.• 3.The value of function at c equals
the value of limit at c.
Another way of saying this is that a function is
continuous at every point in the interval if you can
draw it without lifting your pencil.
ba
c d e
Based on the definition of continuity, which of the functions
are continuous?
discontinuous discontinuous
discontinuous discontinuous continuous
Removable vs. Nonremovable Discontinuities
• A discontinuity at c is called removable if f can be made continuous by appropriately defining (or redefining) f(c).
• If you can simply “plug up the hole”, the discontinuity is removable.
• A discontinuity is nonremovable if there is no way to define the function at a point to make it continuous.
ba
c d e
In the 5 pictures below, let’s now identify which have removable and
nonremovable discontinuities.
nonremovable removable
nonremovable removable continuous