True or false?
If f is continuous at x = c, then f is differentiable at x = c.
not so.
consider y = |x|
Any time there's a cusp, or pointy place, on the graph, f is continuous, but since the slope changes instantly from one value to another, it is not differentiable there.
or, if the slope is vertical, there's no derivative there.
y = ∛x
False.
The statement is false. Just because a function f is continuous at x = c does not guarantee that it is differentiable at x = c. Differentiability at a point requires the existence of a well-defined derivative at that point, which involves the limit of the difference quotient.
To be differentiable at a point, a function must not only be continuous at that point, but it must also have a defined derivative, meaning that the left-hand and right-hand limits of the difference quotient must be equal.
In other words, continuity is a necessary condition for differentiability, but it is not a sufficient condition. So, it is possible to have functions that are continuous but not differentiable at certain points.
False. The statement "If f is continuous at x = c, then f is differentiable at x = c" is not always true. There are cases where a function may be continuous at a certain point, but not differentiable at that point.
To understand this concept, let's first define what it means for a function to be continuous and differentiable at a point:
1. Continuity: A function f is said to be continuous at x = c if three conditions are met:
- f(c) is defined (i.e., f exists at c),
- the limit of f(x) as x approaches c exists, and
- the limit of f(x) as x approaches c is equal to f(c).
2. Differentiability: A function f is said to be differentiable at x = c if the derivative of f exists at c. The derivative measures the rate of change of the function at a particular point.
Now, to determine if the statement is true or false, we need to find counterexamples where a function is continuous at x = c but not differentiable at x = c.
One example of such a function is the absolute value function, f(x) = |x|. The absolute value function is continuous at x = 0 since it is defined and the limit from both sides of 0 exists. However, it is not differentiable at x = 0 because the slope of the tangent line changes abruptly at that point.
Therefore, the statement "If f is continuous at x = c, then f is differentiable at x = c" is false. The continuity of a function does not guarantee its differentiability at a particular point.