Which of the following statements would always be true?
I. If f is differentiable at x = c, then f is continuous at x = c.
II. If f is continuous at x = c, then f is differentiable at x = c.
III. If f is not continuous at x = c, then f is not differentiable at x = c.
Rules
If differentiable, then continuous (hard to define derivative of a function at a point where it jumps)
However if continuous, maybe differentiable maybe not. (example of not is |x| at x = 0
so
1. is true
2. not always
3. is true
logic relation of q 1 to q 3
if a cow, then a mammal hypothesis (assume true)
if a mammal then a cow (false) (converse)
if not a cow, then not a mammal (false)(inverse)
if not a mammal, then not a cow (true)(contrapositive)
see
http://hotmath.com/hotmath_help/topics/converse-inverse-contrapositive.html
The statement that would always be true is I. If f is differentiable at x = c, then f is continuous at x = c.
Explanation:
Statement I is always true because differentiability implies continuity. In other words, if a function is differentiable at a point, it must also be continuous at that point.
Statement II is not always true. There are functions that are continuous at a point but not differentiable at that point. An example is the absolute value function, f(x) = |x|, which is continuous at x = 0 but not differentiable at x = 0.
Statement III is also not always true. There are functions that are not continuous at a point but still differentiable at that point. An example is the function f(x) = 1/x, which is not continuous at x = 0 but is differentiable at x = 0.
To determine which statements would always be true, we need to consider the conditions and implications given in each statement.
I. If f is differentiable at x = c, then f is continuous at x = c.
This statement is true. The differentiability of a function at a point implies its continuity at that point. Therefore, statement I is always true.
II. If f is continuous at x = c, then f is differentiable at x = c.
This statement is not always true. Continuity is a necessary condition for differentiability, but it is not sufficient. There are functions that are continuous but not differentiable at a given point.
III. If f is not continuous at x = c, then f is not differentiable at x = c.
This statement is not always true. Discontinuity at a point does not guarantee non-differentiability. There are certain cases, such as removable discontinuities, where a function can still be differentiable despite being discontinuous at a specific point.
In conclusion, statement I is the only statement that would always be true.