Weistrass function is continuous everywhere but not differentiable everywhere
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Both are polynomials. They are continuous and are differentiable.
Definition: A function f is differentiable at a if f'(a) exists. it is differentiable on an open interval (a, b) [or (a, ∞) or (-∞, a) or (-∞, ∞)]if it is differentiable at every number in the interval.Example: Where is the function f(x) = |x| differentiable?Answer:1. f is differentiable for any x > 0 and x < 0.2. f is not differentiable at x = 0.That's mean that the curve y = |x| has not a tangent at (0, 0).Thus, both continiuty and differentiability are desirable properties for a function to have. These properties are related.Theorem: If f is differentiable at a, then f is continuous at a.The converse theorem is false, that is, there are functions that are continuous but not differentiable. (As we saw at the example above. f(x) = |x| is contionuous at 0, but is not differentiable at 0).The three ways for f not to be differentiable at aare:a) if the graph of a function f has a "corner" or a "kink" in it,b) a discontinuity,c) a vertical tangent
It is; everywhere except at x = 0
The signum function, also known as the sign function, is not differentiable at zero. This is because the derivative of the signum function is not defined at zero due to a sharp corner or discontinuity at that point. In mathematical terms, the signum function has a derivative of zero for all values except at zero, where it is undefined. Therefore, the signum function is not differentiable at zero.
Yes, a polynomial function is always continuous