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Thanks I got c and d. The ones I am having trouble with are a and b.Math_QED said:You can start with c and d. This is an application of the definition of continuity.
A continuous function is a mathematical function that has no abrupt changes or gaps in its graph. This means that the graph of a continuous function can be drawn without lifting your pen from the paper.
A piecewise function is a function that is defined by different rules or equations over different intervals of its domain. This means that the function may have different definitions for different parts of its domain.
The main difference between a continuous function and a piecewise function is that a continuous function is defined by a single rule or equation over its entire domain, while a piecewise function is defined by different rules or equations over different intervals of its domain. This means that a continuous function has a smooth and unbroken graph, while a piecewise function may have abrupt changes or gaps in its graph.
In order for a piecewise function to be continuous, it must have the same value at the points where the different pieces meet. This means that the limit of each piece of the function as it approaches the point of intersection must be equal to the value of the function at that point. Additionally, the function must also be continuous at all other points in its domain.
Yes, a piecewise function can be differentiable at points where the different pieces of the function have the same slope. This means that the function must be continuous at these points and the derivative of each piece must be equal at the point of intersection. However, a piecewise function may not be differentiable at points where the different pieces have different slopes or the function is not continuous.