Continuity Rule
The continuity rule is a fundamental concept in calculus that is used to determine when a function is continuous at a given point or over an interval
The continuity rule is a fundamental concept in calculus that is used to determine when a function is continuous at a given point or over an interval. It helps us understand whether a function has any jumps, breaks, or holes in its graph.
In order for a function to be continuous at a point, three conditions must be satisfied:
1. The function must be defined at that point.
2. The limit of the function as it approaches that point from both the left and the right sides must exist.
3. The limit of the function as it approaches that point from both sides must be equal.
Mathematically, we can express the continuity rule as follows:
If f(x) is a function defined on an interval containing a point c, then f(x) is continuous at c if:
1. f(c) is defined.
2. lim(x->c-) f(x) = lim(x->c+) f(x) = f(c).
Let’s consider an example to understand the application of the continuity rule. Suppose we have the function f(x) = 2x + 3. We want to determine if this function is continuous at x = 2.
First, we check if f(2) is defined. In this case, f(2) = 2(2) + 3 = 7, so it is defined.
Next, we need to evaluate the left-hand limit (lim(x->2-) f(x)) and the right-hand limit (lim(x->2+) f(x)). To find these limits, we substitute values slightly smaller and larger than 2 into the function:
lim(x->2-) f(x) = lim(x->2-) (2x + 3) = 2(2-) + 3 = 4 + 3 = 7
lim(x->2+) f(x) = lim(x->2+) (2x + 3) = 2(2+) + 3 = 4 + 3 = 7
Since both the left and right limits are equal to 7, we can conclude that the limit as x approaches 2 from both sides exists and is equal.
Finally, we compare the limit to f(2). Since lim(x->2-) f(x) = lim(x->2+) f(x) = f(2) = 7, we can say that the function is continuous at x = 2.
Remember, this is just one example of applying the continuity rule. The rule can be used with various types of functions, and it is essential to check for the existence and equality of the limits from both sides.
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