Is the function f defined by f(x)={ x, if x ≤ 1 5, if x>1 . continuous at x=0 ? At

step 1 In the question, we have to prove that the function fx is equals to x to the power n is continuo

Is the function f defined by f(x)={ x, if x ≤ 1 ...

Key Concepts

Continuity at a Point

This concept refers to the idea that a function is continuous at a specific point if the limit of the function as it approaches that point exists and is equal to the function’s value at that point. It requires both the left-hand and right-hand limits at that point to exist and to be identical to the value of the function at that point.

Limits

Limits are the fundamental concept used to analyze the behavior of a function as the input approaches a certain point. They are essential in the study of continuity because they help determine whether the function is 'coming together' at a particular point by comparing the approaching values from both sides.

One-sided Limits

One-sided limits are a specific type of limit that only consider the behavior of the function as the input approaches a point from one direction—either from the left or the right. They are critical for evaluating the continuity of piecewise functions at the points where the function's formula changes.

Piecewise Functions

Piecewise functions are functions that are defined by different expressions over different parts of the domain. Analyzing their continuity requires investigating the limits from each piece at the points where the definition changes, making sure they align with the function's value at those points.

Jump Discontinuity

A jump discontinuity occurs when the left-hand limit and the right-hand limit at a particular point exist but are not equal. This difference creates a 'jump' in the function's graph, indicating that the function is not continuous at that point.

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