Continuity Versus Differentiability

Question

A researcher uses the provided graph of the function $$f(x)= \begin{cases} x+2 & x < 0 \ 2*x+1 & x \ge 0 \end{cases}$$ to evaluate its continuity at $$x = 0$$. The researcher incorrectly concludes that the function is discontinuous at $$x = 0$$ solely because the two pieces have different slopes. What is the error in this reasoning?

Accepted Answer

The error is in confusing continuity with differentiability; continuity at a point only requires that the left-hand and right-hand limits and the function value agree, not that the slopes match.

Answer

The error is in misreading the graph; the graph actually shows that both pieces have the same slope near $$x = 0$$.

Answer

There is no mistake because differing slopes necessarily imply discontinuity at the joining point.

Answer

The error is in the domain specification of the function, which is misrepresented in the graph.