* 10. Let a be a real number and let f be a real-valued function defined on an interval containing x = a. Consider the following conditional statement: If f is differentiable at x =a, then f is continuous at x =a. Which of the following statements have the same meaning as this conditional statement and which ones are negations of this conditional statement? Note: This is not asking which statements are true and which are false. It is asking which statements are logically equivalent to the given statement. It might be helpful to let P represent the hypothesis of the given statement, Q represent the conclusion, and then determine a symbolic representation for each statement. Instead of using truth tables, try to use already established logical equivalencies to justify your conclusions. c080 BY NO SA
* 10. Let a be a real number and let f be a real-valued function defined on an interval containing x = a. Consider the following conditional statement: If f is differentiable at x =a, then f is continuous at x =a. Which of the following statements have the same meaning as this conditional statement and which ones are negations of this conditional statement? Note: This is not asking which statements are true and which are false. It is asking which statements are logically equivalent to the given statement. It might be helpful to let P represent the hypothesis of the given statement, Q represent the conclusion, and then determine a symbolic representation for each statement. Instead of using truth tables, try to use already established logical equivalencies to justify your conclusions. c080 BY NO SA
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Transcribed Image Text:* 10. Let a be a real number and let f be a real-valued function defined on an
interval containing x = a. Consider the following conditional statement:
If f is differentiable at x = a, then f is continuous at x = a.
Which of the following statements have the same meaning as this conditional
statement and which ones are negations of this conditional statement?
Note: This is not asking which statements are true and which are false. It is
asking which statements are logically equivalent to the given statement. It
might be helpful to let P represent the hypothesis of the given statement, Q
represent the conclusion, and then determine a symbolic representation for
each statement. Instead of using truth tables, try to use already established
logical equivalencies to justify your conclusions.
c080
BY NO SA
2.2. Logically Equivalent Statements
51
(a) If f is continuous at x = a, then f is differentiable at x = a.
(b) If f is not differentiable at x = a, then f is not continuous at x = a.
(c) If f is not continuous at x = a, then f is not differentiable at x = a.
(d) f is not differentiable at x = a or f is continuous at x = a.
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