In the figure, f(x) is a function defined in [a,b]. [figure in the original paper] i

Since the original question refers to a figure that is not provided, I will assume the figure depicts a piecewise function $f(x)$ defined on the interval $[a,b]$ with a jump discontinuity at some point $c\in (a,b)$. The function is continuous from the left and right at $c$, but the left-hand limit (LHL) and right-hand limit (RHL) are not equal, indicating a removable discontinuity or a jump discontinuity.

Model Answer

Step 1: Understanding the Function $f(x)$

The function $f(x)$ is defined on the interval $[a,b]$. From the figure (assumed), we observe:

• The function is continuous on $[a,c)$ and $(c,b]$.
• At $x=c$, there is a jump discontinuity because: $$\underset{x\to c^{-}}{\lim }f(x)\ne \underset{x\to c^{+}}{\lim }f(x)$$
• The function may have different expressions on $[a,c]$ and $[c,b]$.

Step 2: Analyzing the Discontinuity

Let:

• $f(c^{-})={\lim }_{x\to c^{-}}f(x)$ (left-hand limit)
• $f(c^{+})={\lim }_{x\to c^{+}}f(x)$ (right-hand limit)

Since $f(x)$ is not defined at $x=c$ (or defined differently), the function has a jump discontinuity at $x=c$.

Step 3: Drawing the Function (Assumed Figure)

Step 4: Conclusion

• The function $f(x)$ is not continuous at $x=c$ because the left-hand limit and right-hand limit are not equal.
• The function is continuous everywhere else in $[a,b]$.
• If $f(c)$ were defined as either $f(c^{-})$ or $f(c^{+})$, the discontinuity could be removable; otherwise, it remains a jump discontinuity.

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Note: Since the original figure was not provided, this answer assumes a typical piecewise function with a jump discontinuity. If the actual figure shows a different behavior (e.g., infinite discontinuity, removable discontinuity), the analysis would adjust accordingly. Always refer to the given figure for precise details.