Question

*Fill in the blank: Suppose f and g be two real functions continuous at a real number c. Then (f + g) is _________.*

1️⃣ continuous
2️⃣ discontinuous
3️⃣ not say anything​

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

Suppose f and g be two real functions continuous at a real number c. Then (f + g) is ________

1. Continuous

2. Discontinuous

3. Not say anything

CONCEPT TO BE IMPLEMENTED

A function f(x) is said to be continuous at a point x = c if

$\displaystyle \sf{\lim_{x \to c + } f(x) =\lim_{x \to c - } f(x) = f(c) }$

EVALUATION

Here it is given that f and g be two real functions continuous at a real number c

Then we have

$\displaystyle \sf{\lim_{x \to c + } f(x) =\lim_{x \to c - } f(x) = f(c) }$

and

$\displaystyle \sf{\lim_{x \to c + } g(x) =\lim_{x \to c - } g(x) = g(c) }$

Now we check whether f + g is continuous at x = c or not

Right hand limit

$\displaystyle \sf{ = \lim_{x \to c + } \bigg[ f(x) + g(x) \bigg]}$

$\displaystyle \sf{ = \lim_{x \to c + } f(x) + \lim_{x \to c + } g(x)}$

$\displaystyle \sf{ = f(c) + g(c)}$

Left hand limit

$\displaystyle \sf{ = \lim_{x \to c - } \bigg[ f(x) + g(x) \bigg]}$

$\displaystyle \sf{ = \lim_{x \to c - } f(x) + \lim_{x \to c - } g(x)}$

$\displaystyle \sf{ = f(c) + g(c)}$

Thus we have

$\displaystyle \sf{ \lim_{x \to c + } \bigg[ f(x) + g(x) \bigg] = \lim_{x \to c - } \bigg[ f(x) + g(x) \bigg] = f(c) + g(c)}$

Thus f + g is continuous at x = c

FINAL ANSWER

Hence the correct option is 1. Continuous

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