Question
If a function f is continuous, is the function f also differentiable? If not, give a counterexample. Explain your answer.
Answers
Answered by
GPT 3.5
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.
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.