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How do you determine if a function is differentiable and continuous?

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How do you determine if a function is differentiable and continuous?

The definition of differentiability is expressed as follows:

  1. f is differentiable on an open interval (a,b) if lim h → 0 f ( c + h ) − f ( c ) h exists for every c in (a,b).
  2. f is differentiable, meaning exists, then f is continuous at c.

How do you determine if a function is differentiable at x A?

A function is said to be differentiable if the derivative of the function exists at all points in its domain. Particularly, if a function f(x) is differentiable at x = a, then f′(a) exists in the domain.

How do you prove that the function f/x is continuous at x 0 but not differentiable at x 0?

To show that f(x)=|x| is continuous at 0 , show that limx→0|x|=|0|=0 . Use ε−δ if required, or use the piecewise definition of absolute value. and limx→0−|x|=limx→0−(−x)=0 . That is, the derivative does not exist at x=0 .

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How do you check if a function is continuous on an interval?

A function is said to be continuous on an interval when the function is defined at every point on that interval and undergoes no interruptions, jumps, or breaks. If some function f(x) satisfies these criteria from x=a to x=b, for example, we say that f(x) is continuous on the interval [a, b].

Is f/x )= x differentiable at x 0?

For differentiability at x = 0. Hence, f(x) is not differentiable at x = 0.

Is X ABS X differentiable at 0?

The left limit does not equal the right limit, and therefore the limit of the difference quotient of f(x) = |x| at x = 0 does not exist. Thus the absolute value function is not differentiable at x = 0.

Is the function f/x )= x continuous at x 0?

f(0) = 2(0)-|0| = 0-0 = 0. Hence f(x) is continuous at x = 0. From the graph, it is clear that f(x) is continuous at x=0.

How do you know when a function is continuous?

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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).

Is 0 continuous differentiable?

Differentiability and continuity It is differentiable everywhere except at the point x = 0, where it makes a sharp turn as it crosses the y-axis. A cusp on the graph of a continuous function. At zero, the function is continuous but not differentiable.

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