Thus, \( f(1) = -4 \) after removing the discontinuity.

["Understanding How ( f(1) = -4 ) Emerges After Removing the Discontinuity in a Function", "Mathematics often reveals surprising connections between seemingly discontinuous behavior and meaningful function values. One such insight shows how carefully analyzing discontinuities—based on limits and function definition—can clarify total function behavior, including precise function values like ( f(1) = -4 ) after removing discontinuities.", "### What Is a Discontinuity in a Function?", "A discontinuity occurs at a point ( x = c ) if the function ( f(x) ) is not continuous there; that is, when:", "[\n\lim_{x \ o c} f(x) <br/>\neq f(c) \quad \ ext{or} \quad \lim_{x \ o c} f(x) \ ext{ does not exist}\n]", "Discontinuities may arise from jumps, infinite discontinuities, or oscillations near ( c ). Removing a discontinuity means redefining ( f(c) ) to match the limit ( \lim_{x \ o c} f(x) ) if the limit exists, effectively “filling the gap” to make ( f ) continuous.", "### The Case of ( f(1) = -4 ) After Removing the Discontinuity", "Consider a function defined piecewise with a removable discontinuity at ( x = 1 ):", "[\nf(x) =\n\begin{cases}\n\frac{x^2 - 5x + 4}{x - 1} & \ ext{if } x <br/>\ne 1 \\n\ ext{undefined (or different value) } & \ ext{if } x = 1\n\end{cases}\n]", "At first glance, plugging in ( x = 1 ) gives:", "[\nf(1) = \frac{1 - 5 + 4}{0} = \frac{0}{0}\n]", "which is indeterminate—not undefined per se, but revealing a point where the function is not directly assigned and may be discontinuous.", "However, let’s simplify the expression for ( x <br/>\ne 1 ):", "Factor the numerator:\n[\nx^2 - 5x + 4 = (x - 1)(x - 4)\n]", "So for ( x <br/>\ne 1 ):", "[\nf(x) = \frac{(x - 1)(x - 4)}{x - 1} = x - 4\n]", "This simplified form ( f(x) = x - 4 ) is valid and continuous everywhere—including at ( x = 1 ).", "### Removing the Discontinuity", "Because ( \lim_{x \ o 1} f(x) = \lim_{x \ o 1} (x - 4) = 1 - 4 = -3 ), we see the limit exists and equals (-3). Yet originally ( f(1) ) may have been defined differently—or undefined—creating a discontinuity.", "Important clarification: While the original function’s value at ( x = 1 ) might be undefined or assigned a different number, the limit exists and equals (-3). Removing the discontinuity means redefining ( f(1) ) to match this limit. Thus:", "[\n\ ext{After redefining } f(1) = -3, \quad f(1) = -4 \quad \ ext{is false unless explicitly reassigned.}\n]", "But in many educational and conceptual contexts, “removing the discontinuity” means setting ( f(1) = \lim_{x \ o 1} f(x) = -3 ), not necessarily (-4). So why does the value become (-4)?", "The key nuance: If the original function was associated with ( f(1) = -4 ) despite the simplification suggesting (-3), then removing the discontinuity means redefining ( f(1) = -3 ) to eliminate the jump—but if instead the original value was truly (-4) and not aligned with the limit, redefining to (-3) removes the discontinuity, but ( f(1) ) becomes (-3), not (-4).", "However, in some interpretations or contexts, manipulating discontinuities involves adjusting values to make functions continuous and consistent—such adjustments may redefine critical points, including ( f(1) ), to reflect true behavior.", "So, if the value ( f(1) = -4 ) was used before removing the discontinuity while theoretically contradicting the limit, removing it properly defines ( f(1) = -3 )—but if ( f(1) ) was moved to (-4) after analysis, this suggests either:", "- An error in initial definition,\n- A reinterpretation, or\n- The value (-4) was a placeholder, and after removing the jump and consistent analogies, we assign ( f(1) = -3 ).", "But suppose the partial simplification or limit process involved a scalar shift—such as adding or subtracting 1 during analysis—and the revised function now satisfies ( \lim_{x \ o 1} f(x) = -4 ), then removing the discontinuity means setting:", "[\nf(1) = -4\n]", "—but only if this value matches the actual limit, which it does not under standard simplification (( x - 4 \ o -3 )).", "Therefore, for ( f(1) = -4 ) to be valid after removing discontinuity, it must result from a corrected or redefined function where:", "- The limit is ( -4 ),\n- Discontinuity exists due to removable jump or mistake in initial assignment,\n- Redefining ( f(1) = -4 ) preserves continuity only if ( \lim_{x \ o 1} f(x) = -4 ) — which must override the ( x - 4 ) cancellation.", "Hence, in context, after removing the discontinuity by consistent reassignment based on limit behavior, and interpreting ( f(1) = -4 ) as the corrected function value, we clarify:", "### The Math Behind It", "Assume ( f(x) = \frac{(x - 1)(x - 4)}{x - 1} ) for ( x <br/>\ne 1 ). Then ( \lim_{x \ o 1} f(x) = -3 ). Removing the discontinuity requires:", "[\n\lim_{x \ o 1} f(x) = f(1)\n\Rightarrow f(1) = -3\n]", "But if educational settings or computations yield ( f(1) = -4 ), this discrepancy must be resolved:", "- Either the original function was numerically or conceptually misrepresented,\n- Or ( f(1) ) was redefined from (-4) to (-3) to reflect continuity and correct function behavior.", "Thus, removing the discontinuity involves choosing ( f(1) ) equal to the limit—so only if the original value diverged legally (e.g., typo, miscalculation), setting ( f(1) = -3 ) resolves it. Saying ( f(1) = -4 ) post-removal implies an inconsistency or restorative adjustment, making (-4) a post-correction label, not the limit.", "### Conclusion", "The statement ( f(1) = -4 ) after removing the discontinuity depends critically on prior definition and analysis. True discontinuity removal requires ( f(1) = \lim_{x \ o 1} f(x) ). If originally ( f(1) = -4 ), but the limit is (-3), then either correction redefines ( f(1) = -3 ), or (-4) reflects an incorrect initial assignment fix. In proper analysis, limits guide continuity; consistent values align with derived limits.", "Thus, when encountering claims like ( f(1) = -4 ) after removal, verify: does the function truly approach (-4)? Or is this a reinterpreted value reflecting reassignment? Independent of ( -4 ), the mathematical principle is clear: continuity after removal means ( f(1) ) equals the limit—regardless of maps or values.", "---", "Key Takeaways:", "- Discontinuities may exist where limits exist but function values differ.\n- Removing discontinuity means remaking ( f(c) ) equal to ( \lim_{x \ o c} f(x) ), if continuous.\n- Value ( f(1) = -4 ) post-removal implies correction from an earlier, inconsistent definition.\n- Always check: Does ( \lim_{x \ o 1} f(x) = -3 )? Then ( f(1) ) must adjust accordingly—even if labeled as (-4) temporarily.", "Understanding this process empowers precise function analysis and accurate mathematical communication—key in calculus, real analysis, and applied mathematics."]