Question

why is a differentiable function continuous?

Answer 1

a function is differentiable at \(x=a\) if\[\lim_{h\to0^-}{f(x+h)-f(x)\over h}=\lim_{h\to0^+}{f(x+h)-f(x)\over h}~~~\text{ at }~~x=a\]a function is continuous if\[\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)\]if there is a discontinuity at \(x=a\)
if the function is not continuous at \(x=a\) either there can be no tangent to that point since it is undefined (removable discontinuity), or the left and right hand derivatives will not be equal (jump discontinuity)

Answer 2

strike the "if there is a discontinuity at \(x=a\)"