Let the functions :$$($$−$$1$$,$$1)$$→ℝ and g:$$($$−$$1$$,$$1)$$→$$($$−$$1$$,$$1)$$ be defined by $$f(x)=$$|$$2x$$−$$1$$|$$+$$|$$2x+1$$| and $$g(x)=x$$−[x] where [x] denotes the greatest integer less than or equal to x. Let f∘g:$$($$−$$1$$,$$1)$$→ℝ be the composite function defined by (f$$∘$$g)(x)=f(g(x)) . Suppose c is the number of points in the interval $$($$−$$1$$,$$1)$$ at which f∘g is NOT continuous, and suppose d is the number of points in the interval $$($$−$$1$$,$$1)$$ at which f∘g is NOT differentiable. Then the value of $$c+d$$ is

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Let the functions :$$($$−$$1$$,$$1)$$→ℝ and g:$$($$−$$1$$,$$1)$$→$$($$−$$1$$,$$1)$$ be defined by $$f(x)=$$|$$2x$$−$$1$$|$$+$$|$$2x+1$$| and $$g(x)=x$$−[x] where [x] denotes the greatest integer less than or equal to x. Let f∘g:$$($$−$$1$$,$$1)$$→ℝ be the composite  function defined by (f$$∘$$g)(x)=f(g(x)) . Suppose c is the number of points in the interval $$($$−$$1$$,$$1)$$ at  which f∘g is NOT continuous, and suppose d is the number of points in the interval $$($$−$$1$$,$$1)$$ at  which f∘g is NOT differentiable. Then the value of $$c+d$$ is

Infinity Learn · Accepted Answer

$$f(x)=$$|$$2x$$−$$1$$|$$+$$|$$2x+1$$|$$g(x)={x}f(g(x))=$$|$$2{x}$$−$$1$$|$$+$$|$$2{x}+1$$| $$=2{x}$$≤$$1/24{x}{x}$$>$$1/2$$ discontinuous at $$x=0$$⇒$$c=1$$ Non differential at $$x=$$−$$1/2$$, $$0$$, $$1/2$$⇒$$d=3$$∴ $$c+d=4$$

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