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evagelos
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Give a formal proof that the function , f(x) = 2x + 3 is continuous over the real Nos R
Continuity is a mathematical concept that refers to the smoothness or connectedness of a function. It means that there are no abrupt changes or gaps in the graph of the function.
Continuity is defined as a property of a function where the limit of the function at a point is equal to the value of the function at that point.
A formal proof of continuity is a rigorous mathematical argument that demonstrates that a function is continuous at a given point. It typically involves the use of the precise definition of continuity and various logical steps to show that the limit of the function at the point in question exists and is equal to the value of the function at that point.
Continuity is important in mathematics because it allows us to make precise statements about the behavior of functions at different points. It is also a fundamental concept in calculus, as continuous functions are necessary for the definition of derivatives and integrals.
To prove continuity using the ε-δ definition, one must show that for any given value of ε (a small positive number), there exists a corresponding value of δ (a small positive number) such that the absolute value of the difference between the input values of the function is less than δ, then the absolute value of the difference between the output values of the function will be less than ε. This shows that the limit of the function at that point is equal to the value of the function at that point, and thus the function is continuous.