Let f :[0,1] rightarrow mathbb{R} and g :[0,1 | vedclass

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(B) Given $f(x) = 1$ if $x \in \mathbb{Q}$ and $f(x) = 0$ if $x 
otin \mathbb{Q}$.Given $g(x) = 0$ if $x \in \mathbb{Q}$ and $g(x) = 1$ if $x 
otin

Vedclass · Accepted Answer

$f + g$ is continuous at the point $x = \frac{2}{3}$ but $f$ and $g$ are discontinuous at $x = \frac{2}{3}$

Vedclass · Answer

(B) Given $f(x) = 1$ if $x \in \mathbb{Q}$ and $f(x) = 0$ if $x 
otin \mathbb{Q}$.Given $g(x) = 0$ if $x \in \mathbb{Q}$ and $g(x) = 1$ if $x 
otin \mathbb{Q}$.Consider the function $h(x) = f(x) + g(x)$.For any $x \in [0,1]$, if $x$ is rational, $h(x) = f(x) + g(x) = 1 + 0 = 1$.If $x$ is irrational, $h(x) = f(x) + g(x) = 0 + 1 = 1$.Thus, $h(x) = 1$ for all $x \in [0,1]$, which is a constant function.$A$ constant function is continuous and differentiable everywhere in its domain.Therefore, $f+g$ is continuous at $x = \frac{2}{3}$.Since $f$ and $g$ are Dirichlet-type functions, they are discontinuous at every point in $[0,1]$.Thus, option $B$ is correct.

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