If a function f is continuous, is the function f also differentiable? If not, give a counterexample. Explain your answer.
No, a function that is continuous is not necessarily differentiable.
A counterexample is the function f(x) = |x|. This function is continuous for all x, as its graph does not have any jumps or breaks. However, it is not differentiable at x = 0.
To see this, consider the definition of differentiability. A function f is differentiable at a point x = c if the limit:
lim┬(h→0)〖(f(c+h)-f(c))/h 〗
exists.
For the function f(x) = |x|, as h approaches 0 from the right (positive values), (f(c+h)-f(c))/h = 1.
Similarly, as h approaches 0 from the left (negative values), (f(c+h)-f(c))/h = -1.
Since the right and left limits are different, the limit as h approaches 0 does not exist. Therefore, f(x) = |x| is not differentiable at x = 0, even though it is continuous.
No, a continuous function is not necessarily differentiable. A counterexample is the function f(x) = |x|.
The function f(x) = |x| is continuous for all real numbers, as it can be drawn without lifting the pen from the paper. However, it is not differentiable at x = 0.
To understand why, we need to examine the left-hand and right-hand derivatives of f(x) at x = 0. The left-hand derivative, denoted f'(0-) or f'(-0), represents the rate of change of the function as x approaches 0 from the left. In this case, as x gets closer to 0 from the left, the function f(x) = |x| behaves like f(x) = -x (since x is negative in this region). Therefore, the left-hand derivative is equal to -1.
On the other hand, the right-hand derivative, denoted f'(0+) or f'(+) represents the rate of change of the function as x approaches 0 from the right. As x gets closer to 0 from the right, f(x) = |x| behaves like f(x) = x (since x is positive in this region). Hence, the right-hand derivative is equal to 1.
Since the left-hand derivative (-1) is not equal to the right-hand derivative (1), the function f(x) = |x| is not differentiable at x = 0.
No, not necessarily. A function can be continuous without being differentiable. To give a counterexample, let's consider the absolute value function, f(x) = |x|. This function is continuous everywhere, meaning that there are no sudden jumps or breaks in the graph. However, it is not differentiable at x = 0.
To explain this, let's consider the definition of differentiability: a function f(x) is differentiable at a point x = a if the derivative of f(x) exists at that point. The derivative measures the rate at which the function is changing, or the slope of the tangent line to the graph at that point.
In the case of f(x) = |x|, we can see that the function has a sharp point at x = 0. The slope of the tangent line to the graph at x = 0 is not well-defined because there are different slopes on either side of the point. One way to see this is to consider the left-hand and right-hand limits.
- For x approaching 0 from the left side, the function f(x) approaches -x, which means the slope of the tangent line is -1.
- For x approaching 0 from the right side, the function f(x) approaches x, which means the slope of the tangent line is +1.
Since the slopes from both sides are different, the derivative does not exist at x = 0. Therefore, even though f(x) = |x| is continuous, it is not differentiable at x = 0.
This counterexample demonstrates that a function can be continuous but not differentiable. It shows that continuity does not guarantee differentiability.