Can a function be continuous and non-differentiable on a given domain??
Yes.
One of the most striking examples of this is the Weierstrass function, discovered by Karl Weierstrass which he defined in his original paper as:
This is a very spiky function that is continuous everywhere on the Real line, but differentiable nowhere.
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Yes, if it has a "bent" point. One example is
where the minus sign means approaching from left and plus sign means approaching from right.
Differentiable function practically means a function that steadily changes its slope (NOT at a constant rate). Therefore, a function that is non-differentiable at a given point practically means that it abruptly changes it's slope from the left of that point to the right.
Let's see 2 functions.
Graph
graph{x^2 [-10, 10, -5.21, 5.21]}
Graph (zoomed)
graph{x^2 [0.282, 3.7, 3.073, 4.783]}
Graph
graph{absx [-10, 10, -5.21, 5.21]}
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Yes, a function can be continuous and non-differentiable on a given domain. An example of such a function is the absolute value function ( f(x) = |x| ). This function is continuous everywhere, including at ( x = 0 ), but it is not differentiable at ( x = 0 ) because it has a sharp corner or "kink" at that point. This means that the derivative of the function does not exist at ( x = 0 ).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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