Show that f(x)=√(x) is continuous at x=0 but not dif. ferentiable at x=0 . Sketch the graph of f

Name: Problem 41, Chapter 3: Calculus A New Horizon
Uploaded: 2021-07-02T21:36:24-08:00
Duration: 1 min 11 s
Channel: Carson Merrill
Description: Show that f(x)=√(x) is continuous at x=0 but not dif. ferentiable at x=0 . Sketch the graph of f

step 1 First problem, we want to show that f of x equals the cubed root of x is continuous, but not dif

Show that f(x)=√(x) is continuous at x=0 but not dif....

Key Concepts

Continuity

Continuity at a point involves ensuring that the limit of a function as the input approaches that point is equal to the function’s value at that point. This concept is crucial for understanding behaviors of functions at specific points and is verified by checking that the left-hand and right-hand limits converge to the same value as the function’s value at that corresponding point.

Differentiability

Differentiability pertains to the existence of a derivative at a point, meaning that the function has a well-defined tangent line there with a finite slope. Even if a function is continuous at a point, it might not be differentiable if the rate of change is inconsistent or if the tangent is vertical, indicating a failure in meeting the conditions required for a derivative.

Limit Definition of the Derivative

The limit definition of the derivative involves calculating the limit of the difference quotient (f(x) - f(a))/(x - a) as x approaches a specific point a. This fundamental concept allows us to understand how the instantaneous rate of change behaves and is essential in analyzing cases where the limit does not yield a finite number, leading to non-differentiability.

Graphical Interpretation of Functions

Graphical interpretation involves drawing the curve of a function based on its behaviors such as continuity and differentiability. In this context, sketching the function helps to visualize how it passes smoothly through certain points while exhibiting features like a vertical tangent or cusp at points of non-differentiability, thus aiding in the overall understanding of the function's behavior.

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