This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To establish the continuity of the function f(z) at z=i, we need to check three conditions: 1. f(i) is defined. 2. _z i f(z) exists. 3. _z i f(z) = f(i). The given function is: f(z) = z^2, & if z ≠ i \\ 0, & if z = i Step 1: Evaluate f(i). According to the definition of the function, when z=i, f(z)=0. f(i) = 0 The function is defined at z=i. Step 2: Evaluate _z i f(z). As z approaches i, z is not equal to i. Therefore, we use the first part of the function definition, f(z) = z^2. _z i f(z) = _z i z^2 Substitute z=i into the expression: _z i z^2 = i^2 = -1 The limit exists and is equal to -1. Step 3: Compare f(i) with _z i f(z). We have f(i) = 0 and _z i f(z) = -1. Since f(i) ≠ _z i f(z), the third condition for continuity is not met. Step 4: Conclude on continuity. The function f(z) is not continuous at z=i. For all other points z ≠ i, f(z) = z^2, which is a polynomial and thus continuous everywhere. Therefore, the function is continuous for all z ≠ i but discontinuous at z=i. The function is discontinuous at z=i. Send me the next one 📸