When providing counter-examples for various things in Calculus, we often utilise piecemeal functions because we can easily 'construct' something 'pathological' by doing that.
Somebody asked me
"To determine if a function is discontinuous or not, can't we just either see if its piecewise defined or if there are any fractions? If the function ISN'T piecewise defined or has any fractions (with x's in the denominator), then wouldn't it always be continuous then?"
I realised that I couldn't think of a counter example immediately, and I still cannot think of one now! Does there exist such a thing at an elementary level?
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