Differentiability implies Continuity

1. Differentiability and Continuity
Earn
your
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2. Section 4­6: Differentiability and Continuity
f(c) exists
exists
A function f(x) is continuous on an interval
if and only if it is continuous at each x­value in the interval.
A function f(x) is continuous
if and only if f(x) is continuous at each x­value in its domain.
Slope of tangent at point x = c.
Instantaneous rate of change at x = c.
A function f(x) is differentiable at a point x = c,
if and only if f '(c) exists.
A function f(x) is differentiable on an interval
if and only if it is differentiable for each x­value in the interval.
A function f(x) is differentiable
if and only if it is differentiable at each x­value in its domain.
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3. Review: Conditional Statements
mortal Definition: subject to death, destined to die.
P Q
Property: If Socrates is a man, then he is mortal.
~ Q ~ P
Contrapositive: If he is not mortal, then Socrates is not a man.
~ P ~ Q
Inverse: If Socrates is not a man, then he is not mortal.
Q P
Converse: If he is mortal, then Socrates is a man.
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4. "smooth"
(Beautiful "S" turns down the slope)
If function f is NOT continuous at x = c,
then function f is NOT differentiable at x = c.
"hang"
tangent
cannot
"hang"
tangent
If function f is NOT differentiable at x = c,
then function f is NOT continuous at x = c.
Not "smooth"
(Switchbacks up the slope)
If function f is continuous at x = c,
then function f is differentiable at x = c.
tangent is vertical at x = c
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5. Cusp
Continuity does NOT imply differentiability
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6. vertical
infinite discontinuity
Continuity does NOT imply differentiability
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7. removable
discontinuity
horizontal
None. No point to "hang" tangent on.
removable discontinuity
If a function is NOT continuous at a point x = c,
then the function is NOT differentiable at the point x = c.
(contrapositive of property)
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