Hey guys, today we're going to talk about continuity and differentiability, two important concepts in calculus. Let's dive in!
Continuity
Continuity is the property of a function where small changes in the input result in small changes in the output. In other words, the function doesn't have any holes or jumps. For example, the function f(x) = x² is continuous because if you make a small change to x, say from 2 to 2.1, the output changes from 4 to 4.41, a small change as well.
Let's take a look at the graph of a continuous function:
As you can see, there are no holes or jumps in the graph.
Continuity is an important property because it allows us to use limit theorems and find derivatives of functions.
Differentiability
Differentiability is the property of a function where its derivative exists at every point in its domain. The derivative is the rate of change of the function, or the slope of the tangent line at any point on the graph of the function.
A function is differentiable if it is continuous and has no sharp corners or cusps. For example, the function g(x) = |x| is continuous but not differentiable at x = 0 because it has a sharp corner at that point.
Let's take a look at the graph of a differentiable function:
As you can see, there are no sharp corners or cusps in the graph.
Differentiability is an important property because it allows us to find maximum and minimum values of a function, which is useful in optimization problems.
In Conclusion
So there you have it, continuity and differentiability are two important concepts in calculus. They allow us to use limit theorems, find derivatives, and optimize functions. Make sure to practice problems and get a good understanding of these concepts for a strong foundation in calculus.
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