check the continuity

let f: Râ†’R
f(x)= x x is irrational
1-x x is rational

5 Answers

1357

Manish Shankar ·2009-07-25 08:24:43

draw the graphs of try to infer from that

how to draw the graph of such function?

1357

Manish Shankar ·2009-07-25 08:31:40

try to draw graphs of function
f(x)=x for all real numbers
f(x)=1-x for all real numbers

and try to think on the question

still not able to do it :(

For any rational number k : f(k) = k

Take any irrational number k+h which is very close to k meaning h tends to 0 but is irrational
f(k+h) = 1-k-h

for f(x) to be continuous : f(k)=f(k+h)
k=1-k-h
k = 0.5 (after neglecting h)

so f(x) is discontinuous at all rational values of x except x = 0.5

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