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What does it mean when a function is not differentiable at a point?
A function is not differentiable at a if its graph has a vertical tangent line at a. The tangent line to the curve becomes steeper as x approaches a until it becomes a vertical line. Since the slope of a vertical line is undefined, the function is not differentiable in this case.
How do you know if a point is not differentiable?
If there derivative can’t be found, or if it’s undefined, then the function isn’t differentiable there. So, for example, if the function has an infinitely steep slope at a particular point, and therefore a vertical tangent line there, then the derivative at that point is undefined.
Does a function have to be defined at a point to be differentiable?
In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.
How do you know if a function is continuous at a point?
Saying a function f is continuous when x=c is the same as saying that the function’s two-side limit at x=c exists and is equal to f(c).
Can derivatives be zero?
The derivative f'(x) is the rate of change of the value of function relative to the change of x. So f'(x0) = 0 means that function f(x) is almost constant around the value x0. Having a derivative means that a function can change only gradually.
How do you know if a function is continuous without graphing?
Your pre-calculus teacher will tell you that three things have to be true for a function to be continuous at some value c in its domain:
- f(c) must be defined.
- The limit of the function as x approaches the value c must exist.
- The function’s value at c and the limit as x approaches c must be the same.
Can a function be differentiable but not continuous?
We see that if a function is differentiable at a point, then it must be continuous at that point. If is not continuous at , then is not differentiable at . Thus from the theorem above, we see that all differentiable functions on are continuous on .