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Homework Statement
Prove that if f is continuous at a, then so is |f|
Homework Equations
The Attempt at a Solution
I know
lim f = L
x->a
Not sure really where to go from here.
A continuous function is a type of mathematical function where the output value changes smoothly and gradually as the input value changes. This means that there are no sudden jumps or breaks in the function's graph.
To prove that a function is continuous, you must show that it satisfies the three conditions of continuity: 1) the function is defined at the point in question, 2) the limit of the function at that point exists, and 3) the limit of the function at that point is equal to the function's value at that point.
Proving continuity is important because it allows us to make accurate predictions and calculations based on a function's behavior. It also helps us understand the behavior of a function and its relationship to other mathematical concepts.
Some common techniques used to prove continuity include using the definition of continuity, using algebraic manipulations, and using theorems such as the Intermediate Value Theorem and the Squeeze Theorem.
No, not all continuous functions are differentiable. While all differentiable functions are continuous, the opposite is not always true. A function can be continuous but not differentiable if it has a sharp point or corner in its graph, known as a point of non-differentiability.