Let f(x+y) = f(x) + f(y) for all x and...

MathematicsContinuity at a Point and in an IntervalJEE Advanced 1981Easy

Let $f (x + y) = f (x) + f (y)$ for all $x$ and $y$ . If the function $f (x)$ is continuous at $x = 0$ , then show that $f (x)$ is continuous at all $x$ .

Visualized Solution (English)

The Functional Equation

Given: $f (x + y) = f (x) + f (y)$ for all real numbers $x$ and $y$ .
This is a fundamental property known as Cauchy's Functional Equation.

Finding the Value at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x = 0

Substitute $x = 0$ and $y = 0$ into the equation.
Calculation: $f (0 + 0) = f (0) + f (0)$
Simplifies to: $f (0) = 2 f (0)$
Therefore, $f (0) = 0$ .

Continuity at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x = 0

$f (x)$ is continuous at $x = 0$ .
By definition: $lim_{h \to 0} f (0 + h) = f (0)$
Since $f (0) = 0$ , we have $lim_{h \to 0} f (h) = 0$ .

General Continuity at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x

To prove continuity at any point $x$ , we check the limit:
$lim_{h \to 0} f (x + h)$

Applying the Property

Using the functional equation: $f (x + h) = f (x) + f (h)$
Substitute this into the limit expression: $lim_{h \to 0} [f (x) + f (h)]$

Evaluating the Limit

$lim_{h \to 0} f (x + h) = f (x) + lim_{h \to 0} f (h)$
Substituting $lim_{h \to 0} f (h) = 0$ :
$lim_{h \to 0} f (x + h) = f (x) + 0 = f (x)$

Conclusion

Since $lim_{h \to 0} f (x + h) = f (x)$ , the function is continuous at all $x$ .
Key Takeaway: Local continuity plus Cauchy's equation implies global continuity.
Next Challenge: What if the equation was $f (x + y) = f (x) f (y)$ ?

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