Continuity And Differentiability

In summary, continuity is the property of a function where there are no sudden changes or breaks in the graph, while differentiability refers to the existence of a derivative at a point. A removable discontinuity occurs when the limit of a function exists, but the function is not defined at that point. To determine if a function is differentiable, you can use the limit definition of the derivative. Continuity and differentiability are both necessary for a function to be considered smooth.
  • #1
himanshu121
653
1
Consider [tex]f(x)=x^3-x^2+x+1 [/tex]
[tex] g(x)=\left\{\begin{array}{cc}{max\{f(t),0\leq t \leq x\}}\;\ 0\leq x \leq 1
\\ 3-x\;\ 1< x \leq 2\end{array}\right[/tex]

Discuss the continuity and differentiability of g(x) in the interval (0,2)

I know how to do it
As f(x) is increasing function therefore max will be x^3-x^2+x+1.
But
I want to know the problem graphically ??[?]
 
Last edited:
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  • #2
Oh! I am also in difficulty with "Continuity And Differentiability" like limits! Seems you can explain a little :)
 
  • #3


The graph of f(x)=x^3-x^2+x+1 is a cubic function with a positive leading coefficient, meaning it is an upward facing parabola. The graph is continuous and differentiable for all values of x.

The graph of g(x) is a piecewise function, with the first piece being the same as f(x) and the second piece being a straight line with a negative slope.

In the interval (0,1), g(x) will be equal to f(x) and will have the same continuity and differentiability as f(x).

In the interval (1,2), g(x) will be equal to 3-x, which is a continuous and differentiable function.

Therefore, g(x) is continuous and differentiable in the interval (0,2) as it is made up of continuous and differentiable functions in each subinterval.

Graphically, this can be seen as a smooth curve for the first part of the graph (0,1) and a straight line for the second part (1,2). The graph will have no breaks or sharp turns, indicating continuity, and it will have a smooth slope throughout, indicating differentiability.
 

FAQ: Continuity And Differentiability

1. What is the definition of continuity?

The definition of continuity is the property of a function where there are no sudden changes or breaks in the graph. This means that the function is continuous at every point in its domain.

2. How is continuity different from differentiability?

Continuity and differentiability are related but different concepts. Continuity focuses on the smoothness of a function, while differentiability refers to the existence of a derivative at a point. A function can be continuous but not differentiable, but if a function is differentiable, it must also be continuous.

3. What is a removable discontinuity?

A removable discontinuity occurs when there is a break in the graph of a function, but the function can be redefined at that point to make it continuous. This means that the limit of the function exists, but the function is not defined at that point.

4. How can you determine if a function is differentiable?

A function is differentiable if the derivative exists at every point in its domain. This means that the function must be continuous and have a well-defined slope at every point. To determine if a function is differentiable, you can use the limit definition of the derivative to check if the limit exists at each point.

5. What is the relationship between continuity, differentiability, and smoothness?

Continuity and differentiability are both necessary for a function to be considered smooth. This means that a function is continuous and has a well-defined derivative at every point, making it smooth and without any sudden changes or breaks in the graph.

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